Webmaster The language of variation is often used to describe various aspects of the

An _______ is a statement indicating that two expressions are equal.

2x  1  4 is an example of a _______ equation in one variable.

A number that makes an equation true when substituted for the variable is called a ______ .

If two equations have the same solution set, they are called ______ equations.

An equation that is made true by any permissible replacement value for the variable is called an ______ .

An equation that is false for all replacement values for the variable is called a _______.

<, >, and  are _____ symbols.

To _____ an inequality means to find all values of the variable that make the inequality true.

a. Adding the _____ number to, or subtracting the same number from, both sides of an equation does not change its solution. b. Multiplying or dividing both sides of an equation by the _____ nonzero number does not change its solution.

If we multiply both sides of an inequality by a negative number, the direction of the inequality must be ______ for the inequalities to have the same solutions.

Use a check to determine whether is a solution of the following equation and inequality. a. 5(2x  7)  2x 4 b. 3x  6 95

The solution set of a linear inequality in is graphed on the right. Determine whether a true or false statement results when a. 4is substituted for x. b. 3 is substituted for x c. 0 is substituted for x. 4 2

Match each interval with its graph. a. ( 1 ) b. (1) c (1) i. 0 2 ii. 2 0 iii. 2 0

a. Suppose that when solving a linear equation, the variable drops out, and the result is 7  1. What is the solution set? b. Suppose that when solving a linear inequality, the variable drops out, and the result is 6 10. Write the solution set in interval notation and graph it.

Solve each equation. Check the result. See Example 1. 4x  1  13

Solve each equation. Check the result. See Example 1. 4x 8  16

Solve each equation. Check the result. See Example 1. 3(x  1)  15

Solve each equation. Check the result. See Example 1. 2(x  5)  30

Solve each equation. Check the result. See Example 1. 2x  6(2x  3)  10 3

Solve each equation. Check the result. See Example 1. 3(2y 4) 6  3y 3

Solve each equation. Check the result. See Example 1. 7(a  2)  11a  17 7a 5

Solve each equation. Check the result. See Example 1. 5(5 a)  4a  37 6a 2

Solve each equation. Check the result. See Example 2. 1 2 x 4  1  2x 7

Solve each equation. Check the result. See Example 2. 2x  3  2 3 x 1 1

Solve each equation. Check the result. See Example 2. x 2 x 3  4

Solve each equation. Check the result. See Example 2. x 2  x 3  10

Solve each equation. Check the result. See Example 2. 1 6 (x  12)  1  x 3

Solve each equation. Check the result. See Example 2. 3 2 (y  4)  20 y 2

Solve each equation. Check the result. See Example 2. 5 (x  6)  5 8 (2x 1)  2

Solve each equation. Check the result. See Example 2. 3 (x 3)  1 6 (7x  29)  3

Solve each equation. If an equation is an identity or a contradiction, so indicate. See Example 3. 2x 6  2x  4(x 2) 2

Solve each equation. If an equation is an identity or a contradiction, so indicate. See Example 3. 3x  2x  1 (5  x) 2x

Solve each equation. If an equation is an identity or a contradiction, so indicate. See Example 3. 2y  1  5(0.2y  1) (4 y) 3

Solve each equation. If an equation is an identity or a contradiction, so indicate. See Example 3. 4(2 3t)  6t  6t  8 2y

Solve each equation. If an equation is an identity or a contradiction, so indicate. See Example 3. 7 2 (y 1)  1 2  1 2 (7y 6) 4

Solve each equation. If an equation is an identity or a contradiction, so indicate. See Example 3. 2(x 3)  3 2 (x 4)  x 2 7

Solve each equation. If an equation is an identity or a contradiction, so indicate. See Example 3. 0.3(x 4)  0.6  0.2(x  4)  0.5x 2

Solve each equation. If an equation is an identity or a contradiction, so indicate. See Example 3. . 0.5(y  2)  0.7 0.3y  0.2(y  9)

Solve each formula for the specified variable. See Examples 4 and 5. P  2l  2w for w

Solve each formula for the specified variable. See Examples 4 and 5. P  2l  2w l

Solve each formula for the specified variable. See Examples 4 and 5. V  B 1 3 B

Solve each formula for the specified variable. See Examples 4 and 5. A  b 1 2  B bh

Solve each formula for the specified variable. See Examples 4 and 5. T W  ma W

Solve each formula for the specified variable. See Examples 4 and 5. G  U TS  PV S

Solve each formula for the specified variable. See Examples 4 and 5. z  x i x m s

Solve each formula for the specified variable. See Examples 4 and 5. P  L  s

Solve each formula for the specified variable. See Examples 4 and 5. S  l n(a  l) 2

Solve each formula for the specified variable. See Examples 4 and 5. h  48t  a 1 2 at2

Solve each formula for the specified variable. See Examples 4 and 5. l  a  (n 1)d d

Solve each formula for the specified variable. See Examples 4 and 5. P  2(w  h  l) h

Solve each inequality. Graph the solution set and write it in interval notation. See Example 6. 5x 3  7 7

Solve each inequality. Graph the solution set and write it in interval notation. See Example 6. 7x 9  5

Solve each inequality. Graph the solution set and write it in interval notation. See Example 6. 9a  11 29 3

Solve each inequality. Graph the solution set and write it in interval notation. See Example 6. 3b 26 4

Solve each inequality. Graph the solution set and write it in interval notation. See Example 6. 3(z 2) 2(z  7) 5(

Solve each inequality. Graph the solution set and write it in interval notation. See Example 6. 5(3  z)  3(z  3)

Solve each inequality. Graph the solution set and write it in interval notation. See Example 6. 2x  4  6x  2 3x  2 5

Solve each inequality. Graph the solution set and write it in interval notation. See Example 6. 5x  6  2x 2 x  4

Solve each inequality. Write the solution set in interval notation and then graph it. See Example 7. 3x 1 5 2x

Solve each inequality. Write the solution set in interval notation and then graph it. See Example 7. 2x  6 16

Solve each inequality. Write the solution set in interval notation and then graph it. See Example 7. 5t  3 5 9t

Solve each inequality. Write the solution set in interval notation and then graph it. See Example 7. 9t  6 16

Solve each inequality. Write the solution set in interval notation and then graph it. See Example 7. 7y  5  5y 1 8

Solve each inequality. Write the solution set in interval notation and then graph it. See Example 7. 8 9y y 5

Solve each inequality. Write the solution set in interval notation and then graph it. See Example 7. t  1 3t t 20 a 

Solve each inequality. Write the solution set in interval notation and then graph it. See Example 7. a  4 10a  a 16 7

Solve each inequality. Graph the solution set and write it in interval notation. See Example 8 and the Success Tip in the margin. 2(5x 6)  4x 15  6x

Solve each inequality. Graph the solution set and write it in interval notation. See Example 8 and the Success Tip in the margin. 3(4x 2)  14x 7 2x

Solve each inequality. Graph the solution set and write it in interval notation. See Example 8 and the Success Tip in the margin. 3b  7 3 2b 9 2 3

Solve each inequality. Graph the solution set and write it in interval notation. See Example 8 and the Success Tip in the margin. 5x 4  3 5x 4

Solve each equation. If an equation is an identity or a contradiction, so indicate. 2r 5  1 r 5s

Solve each equation. If an equation is an identity or a contradiction, so indicate. 5s 13  s 1 5

Solve each equation. If an equation is an identity or a contradiction, so indicate. 0.2(a 5) 0.1(3a  1)  0 2

Solve each equation. If an equation is an identity or a contradiction, so indicate. 0.8(3a 5) 0.4(2a  3)  1.2

Solve each equation. If an equation is an identity or a contradiction, so indicate. 4 5 s  2

Solve each equation. If an equation is an identity or a contradiction, so indicate. 9 8 s  3

Solve each equation. If an equation is an identity or a contradiction, so indicate. 1 2 (3y  2) 5 8  3 4 y

Solve each equation. If an equation is an identity or a contradiction, so indicate. 3 4 (4c 3)  7 8 c  19 16

Solve each equation. If an equation is an identity or a contradiction, so indicate. 8x  3(2 x)  5x  6 3

Solve each equation. If an equation is an identity or a contradiction, so indicate. 2(2a  1) 1  4a  1

Solve each equation. If an equation is an identity or a contradiction, so indicate. 12  3(x 4) 21  5[5 4(4 x)] 2(2

Solve each equation. If an equation is an identity or a contradiction, so indicate.  3[2  6(4 2x)]  (x  3) 12

Solve each equation. If an equation is an identity or a contradiction, so indicate. 3  p 3 4p  1 p  7 2 1

Solve each equation. If an equation is an identity or a contradiction, so indicate. 4 t 2 3t 5  2  t  1 3 3

Solve each equation. If an equation is an identity or a contradiction, so indicate. 5x  10  5x

Solve each equation. If an equation is an identity or a contradiction, so indicate. 4(t 2) t  (9 3t) 5x

Solve each equation. If an equation is an identity or a contradiction, so indicate. 0.06(a  200)  0.1a  172

Solve each equation. If an equation is an identity or a contradiction, so indicate. 0.03x  0.05(6,000 x)  280

Solve each equation. If an equation is an identity or a contradiction, so indicate. 4[p (3 p)]  3(6p 2) 0.0

Solve each equation. If an equation is an identity or a contradiction, so indicate. 2[5(4 a)  2(a 1)]  3 a 4[

Solve each inequality. Graph the solution set and write it in interval notation. 3(a  2)  2(a  1) 4

Solve each inequality. Graph the solution set and write it in interval notation. 4(y 1)  y  8 2

Solve each inequality. Graph the solution set and write it in interval notation. x 7 2 x 1 5 x 4 3(a

Solve each inequality. Graph the solution set and write it in interval notation. 3a  1 3 4 3a 5 1 15 x

Solve each inequality. Graph the solution set and write it in interval notation. 5(2n  2) n  3n 3(1 2n) 3a

Solve each inequality. Graph the solution set and write it in interval notation. 1  4(y 1)  2y 1 2 (12y 30)  15 5(2

Solve each inequality. Graph the solution set and write it in interval notation. 0.4x  0.4 0.1x  0.85 0

Solve each inequality. Graph the solution set and write it in interval notation. 0.05  0.8x 0.5x 0.7 1

Solve each inequality. Graph the solution set and write it in interval notation. 1 2 y  2 1 3 y 4

Solve each inequality. Graph the solution set and write it in interval notation. 4 x 1 3 x  2 1

Solve each inequality. Graph the solution set and write it in interval notation. 7  a 9 5 3 a 3 1

Solve each inequality. Graph the solution set and write it in interval notation. 5  7 2 7  a 9

Simplify each expression and solve each equation. a. 2 (6x  8) 10 2 3 (6x 9) 5 b. 2 (6x  8) 10  2 3 (6x 9) 1

Simplify each expression and solve each equation. a. 6.31w  9.22  5(7.21w 1.13) b. 6.31w  9.22  5(7.21w 1.13)

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. 12x 33.16 5.84 b. 12x 33.16  5.84

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your answer to part a to determine the solution set for the inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. 3 4 x  21 32 b. 3 4 x 21 32 3

Spring Tours. A group of junior high students will be touring Washington, D.C. Their chaperons will have the $1,810 cost of the tour reduced by $15.50 for each student they supervise. How many students will a chaperon have to supervise so that his or her cost to take the tour will be $1,500?

Machining. Each pass through a lumber plane shaves off 0.015 inch of thickness from a board. How many times must a board, originally 0.875 inch thick, be run through the planer if a board of thickness 0.74 inch is desired?

Moving Expenses. To help move his furniture, a man rents a truck for $41.50 per day plus per mile. If he has budgeted $150 for transportation expenses, how many miles will he be able to drive the truck if the move takes 1 day?

Computing Salaries. A student working for a delivery company earns $57.50 per day plus $4.75 for each package she delivers. How many deliveries must she make each day to earn $200 a day?

Fencing Pens. A man has 150 feet of fencing to build the two-part pen shown in the illustration. If one end is a square and the other a rectangle, find the outside dimensions of the pen. x ft (x + 5) ft Square Rectangle

Fencing Pastures. A farmer has 624 feet of fencing to enclose a pasture. Because a river runs along one side, fencing will be needed on only three sides. Find the dimensions of the pasture if its length is double its width. Width

Webmaster One of the most important duties that a webmaster has is to monitor the web traffic of the site that he or she oversees. Suppose a counter for a news organization website has recorded 650,568,999 views for April 1 through April 29. How many views does the site need on the last day of the month (April 30) to average 22,000,000 daily visitors?

Averaging Grades. A student has scores of 70, 77, and 85 on three government exams. What score does she need on a fourth exam to give her an average of 80 or better?

Fundraising. A school PTA wants to rent a dunking tank for its annual school fundraising carnival. The cost is $85.00 for the first 3 hours and then $19.50 for each additional hour or part thereof. How long can the tank be rented if up to $185 is budgeted for this expense?

Work Schedules. A student works two part-time jobs. He earns $8 an hour for working at the college library and $15 an hour for construction work. To save time for study, he limits his work to 25 hours a week. If he enjoys the work at the library more, how many hours can he work at the library and still earn at least $300 a week?

Scheduling Equipment. An excavating company charges $300 an hour for the use of a backhoe and $500 an hour for the use of a bulldozer. (Part of an hour counts as a full hour.) The company employs one operator for 40 hours per week to operate the machinery. If the company wants to bring in at least $18,500 each week from equipment rental, how many hours per week can it schedule the operator to use a backhoe?

Video Game Systems. A student who can afford to spend up to $1,000 sees the ad shown in the illustration. If she decides to buy the video game system, find the greatest number of video games that she also can purchase. (Disregard sales tax.) VIDEO GAME SYSTEM only $44999 YOUR FAVORITE GAMES only $4599 each

Explain why the equation x  x  1 doesnt have a real-number solution.

Explain what is wrong with the following statement: When solving inequalities involving negative numbers, the direction of the inequality symbol must be reversed.

Simplify each expression. Write answers using only positive exponents. a t 3 t 5 t 6 t 2 t 4 b 3 x

Simplify each expression. Write answers using only positive exponents. a a2 b3 a5 b2 a6 b5 b 4 a

Find the value of that makes 4 a solution of the following linear equation in x. k  3x 6  3kx k  16 x

Solve: 0.75(x 5) 4 5  1 6 (3x  1)  3.2 k

Consider the following solution of the inequality 1 3  1 x where it appears that the solution set is the interval (3, ) 3 3xa 1 3 b  3xa 1 x b 1 a. Show that x  1 makes the original inequality true. b. If x  1 makes the original inequality true, there must be an error in the solution. Where is it?

Medical Plans. A college provides its employees with a choice of the two medical plans shown in the following table. For what size hospital bills is Plan 2 better for the employee than Plan 1? (Hint: The cost to the employee includes both the deductible payment and the employees coinsurance payment.) Plan 1 Plan 2 Employee pays $100 Employee pays $200 Plan pays 70% of the rest Plan pays 80% of the rest

A set of ordered pairs is called a ______. The set of all first components of the ordered pairs is called the _____ and the set of all second components is called the .

A ______ is a set of ordered pairs (a relation) in which to each first component there corresponds exactly one second component.

Given a relation in x and y, if to each value of x in the domain there corresponds exactly one value of y in the range, y is said to be a______ of x. We call x the independent ______ and y the _____ variable.

For a function, the set of all possible values that can be used for the independent variable is called the _____ . The set of all values of the dependent variable is called the _____ .

A _____ function is a function that can be defined by an equation of the form (x)  mx  b. A polynomial function is a function whose equation is defined by a polynomial in _____variable.

We call (x)  x the _____ function because it assigns each real number to itself. We call f(x)  2 a _____ function, because for any input , the output is always 2.

U.S. Recycling. The following table gives the approximate number of aluminum cans (in billions) collected each year for the years 20002006. a. Display the data in the table as a relation, that is, as a set of ordered pairs. b. Find the domain and range of the relation. c. Use an arrow diagram to show how members of the range correspond to members of the domain. Year 2000 2001 2002 2003 2004 2005 2006 Billions of 63 56 54 50 52 51 51 aluminum cans

For the given input, what value will the function machine output? 5 ? f(x) = x3 x

Explain why is not in the domain of (x)  1 x  4 4

Consider the linear function y  4 5x  3. a. What is the slope of its graph? b. What is the y-intercept of its graph?

Consider the linear function f(x)  6x 4 y. a. What is the y-intercept of its graph? b. What is the x-intercept of its graph?

The graphing calculator display shows a table of values for a function . Fill in the blanks: (1)  (3) 

Fill in the blanks. a. We read (x)  5x 6 as is equal to minus 6. b. We read g(t)  t  9 as is equal to plus 9. g

Fill in the blanks.This variable represents the (x)  2x 5. This is the Use this expression of the function. to find the

Fill in the blank so that the following statements are equivalent: If y  5x  1, find the value of y when x  8 . If (x)  5x  1, find

Fill in the blanks. If (2)  7, the input 2 and the output 7 can be written as the ordered pair

Fill in the blanks. When graphing (x)  x  5, the vertical axis of the rectangular coordinate system can be labeled or

Fill in the blanks. To write the slopeintercept form y  mx  b and the pointslope form y y1  m(x x1)y using function notation, we simply replace y with .

Find the domain and range of each relation. See Example 1. {(7, 1), (1, 11), (5, 3), (8, 6)}

Find the domain and range of each relation. See Example 1. {(15, 3), (0, 0), (4, 6), (3, 8)}

Find the domain and range of each relation. See Example 1. {(0, 1), (23, 35), (7, 1)}

Find the domain and range of each relation. See Example 1. {(1, 12), (6, 8), (5, 8), (0, 0), (1, 4)}

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. 10 20 30 20 40 60

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. 4 2 0 6 8 10 12

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. 4 2 4 6

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. 5 10 15 15

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. {(3, 4), (3, 4), (4, 3), (4, 3)} 5

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. {(1, 1), (3, 1), (5, 1), (7, 1), (9, 1)} {(3,

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. {(2, 7), (1, 10), (0, 13), (1, 16)}

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. {(2, 4), (3, 8), (3, 12), (4, 16)} {(2

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. 1 7 2 15 3 23 4 16 5 8 yx

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. 30 2 30 4 30 6 30 8 30 10 yx

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. 6 0 0 2 4 1 2 3 1 4 yx 1 1

Determine whether the relation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 2. 1 1 2 1 3 1 4 1 yx

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. y  2x  3

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. y  4x 1 0

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. y  4x  x 2

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. y2  x

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. y 4  x

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. y  1 x

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. xy  9

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. y  0 x 0

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. y  1 x 2

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. x  1  0 y 0

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. x  0 y 0

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 3. xy  4

Find and . See Example 4. (x)  3x

Find and . See Example 4. (x)  4x

Find and . See Example 4. (x)  2x 3

Find and . See Example 4. (x)  3x 5

Find and . See Example 4. g(x)  x 2 2 10 g

Find and . See Example 4. g(x)  x 2 g(x)  x 2 2

Find and . See Example 4. g(x)  x x 3  x g

Find and . See Example 4. g(x)  x3 g(x)  x x 3

Find and . See Example 4. g(x)  (x  1)2

Find and . See Example 4. g(x)  (x 3)2 g

Find and . See Example 4. g(x)  2x  2x  2 2 x  1 g

Find and . See Example 4. g(x)  5x2 g(x)  2x  2x  2

Find and . See Example 4. h(x)  0 x 0  2

Find and . See Example 4. h(x)  0 x 0 5 h

Find and . See Example 4. h(x)  1 x  3

Find and . See Example 4. h(x)  3 x 4

Find and . See Example 4. h(x)  x x 3

Find and . See Example 4. h(x)  x x 2  2

Find and . See Example 4. h(x)  x2  2x 35 x2  5x  6

Find and . See Example 4. h(x)  x2  x 2 x2 5x h(

Complete each table. See Example 4. (t)  0 t 2 0  9 5.4 1.7 t f

Complete each table. See Example 4. (r)  2r 2 0  1 h( Input Output 0.9 5.4 1.7

Complete each table. See Example 4. g(a)  a 2 3 Input Output 5 2 1 6 3 4 5

Complete each table. See Example 4. g(b)  21b 1 4 g( b gb 5 2 1 6 3 4

Find and . See Example 4. g(x)  2x

Find and . See Example 4. g(x)  3x

Find and . See Example 4. g(x)  3x 5 g

Find and . See Example 4. g(x)  2x 7

Let f(x)  2x  5. For what value of x does function f have the given value? See Example 5. (x)  5

Let f(x)  2x  5. For what value of x does function f have the given value? See Example 5. (x)  7

Let (x)  3 2 x 2. For what value of x does function f have the given value? See Example 5. (x)  1 2

Let (x)  3 2 x 2. For what value of x does function f have the given value? See Example 5. (x)  2 3

Find the domain of each function. See Example 6. a. h(x)  3x  6 b. (x)  1 x 4

Find the domain of each function. See Example 6. a. g(x)  0 x 7 0 b. (x)  5 x  1

Find the domain of each function. See Example 6. a. (x)  x 2 b. s(x)  9 2x  1

Find the domain of each function. See Example 6. a. h(x)  x3 b. t(x)  15 1 3x

Graph each function. See Objective 5. (x)  2x 1

Graph each function. See Objective 5. (x)  x  2

Graph each function. See Objective 5. (x)  x 2 3 2 x 3 (

Graph each function. See Objective 5. (x)  2 3 (x)  x 2 3

Graph each function. See Objective 5. (x)  x

Graph each function. See Objective 5. (x)  x

Graph each function. See Objective 5. (x)  4

Graph each function. See Objective 5. (x)  2

Graph each function. See Objective 5. g(x)  0.75x

Graph each function. See Objective 5. g(x)  0.25x

Graph each function. See Objective 5. s(x)  x  3 7 8 x  2

Graph each function. See Objective 5. s(x)  4 5 s(x)  x  3

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope 5, y-intercept (0, 3)

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope 2, y-intercept (0, 11)

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope , passes through (10, 1)

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Slope , passes through (8, 1)

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through (1, 7) and (2, 1)

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through (2, 2) and (2, 8) 1

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through (3, 0), parallel to the graph of g(x)  2 3 x 4 1

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through (2, 20), parallel to the graph of g(x)  8x  1

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through (1, 2), perpendicular to the graph of g(x)  x 6  1 g

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through (54, 0), perpendicular to the graph of g(x)  9x  5

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Horizontal, passes through (8, 12)

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Horizontal, passes through (9, 32)

Webmaster According to data from Netcraft Web Server Survey, the total number of active websites available over the Internet (in millions) is approximated by the function , where t is the number of years after 2005. Approximately how many active websites were there in 2011?

Decongestants. The temperature in degrees Celsius that is equivalent to a temperature in degrees Fahrenheit is given by the linear function C(F)  5 9 (F 32). Refer to the label from a bottle of decongestant shown below. Use this function to find the low and high temperature extremes, in degrees Celsius, in which the bottle should be stored.

Concessionaires. A baseball club pays a vendor $125 per game for selling bags of peanuts for $4.75 each. a. Write a linear function that describes the profit the vendor makes for the baseball club during a game if she sells bags of peanuts. b. Find the profit the baseball club will make if the vendor sells 110 bags of peanuts during a game.

Home Construction. In a proposal to some clients, a housing contractor listed the following costs: a. Write a linear function that the clients could use to determine the cost of building a home having square feet. b. Find the cost to build a home having 1,950 square feet. Fees, permits, miscellaneous Construction, per square foot $12,000 $95

Nurses. The demand for full-time registered nurses in the United States can be modeled by a linear function. In 2005, approximately 2,175,500 nurses were needed. By the year 2015, that number is expected to increase to about 2,586,500. (Source: National Center for Health Workforce Analysis) a. Let t be the number of years after 2000 and N be the number of full-time registered nurses needed in the U.S. Write a linear function N(t) to model the demand for nurses. b. Use your answer to part a to predict the number of full-time registered nurses that will be needed in 2025, if the trend continues.

Wood Production. The total world wood production can be modeled by a linear function. In 1960, approximately 2,400 million cubic feet of wood were produced. Since then, the amount of increase has been approximately 25.5 million cubic feet per year. (Source: Earth Policy Institute) a. Let t be the number of years after 1960 and W be the number of million cubic feet of wood produced. Write a linear function W(t) to model the production of wood. b. Use your answer to part a to estimate how many million cubic feet of wood the world produced in 2010.

Breathing Capacity. When fitness instructors prescribe exercise workouts for elderly patients, they must take into account age-related loss of lung function. Studies show that the percent of remaining breathing capacity for someone over 30 years old can be modeled by a linear function. (Source: alsearsmd.com) a. At 35 years of age, approximately 90% of maximal breathing capacity remains and at 55 years of age, approximately 66% of maximal breathing capacity remains. Let a be the age of a patient and L be the percent of her maximal breathing capacity that remains. Write a linear function L(a) to model this situation. b. Use your answer to part a to estimate the percent of maximal breathing capacity that remains in an 80-year-old.

Chemical Reactions. When students mixed solutions of acetone and chloroform, they found that heat was generated. However, as time passed, the mixture cooled down. The graph shows data points of the form (time, temperature) taken by the students. a. The linear function models the relationship between the elapsed time since the solutions were combined and the temperature of the mixture. Graph the function. b. Predict the temperature of the mixture immediately after the two solutions are combined. c. Is more or less than the temperature recorded by the students for ? 28 29 30 31 600 120 180 240 300 360 Elapsed time (sec) Temperature (C) Initial temperatures: Acetone Chloroform}23.6C T(

Taxes. The function T(a) 837.50 0.15(a  8,375) (where is adjusted gross income) is a model of the instructions given on the first line of the following tax rate Schedule X. a. Find and interpret the result. b. Write a function that models the second line on Schedule X. Schedule XUse if your filing status is Single If your adjusted gross income is: Over $ 8,375 $34,000 $34,000 $82,400 $ 837.50 + 15% $4,681.25 + 25% $ 8,375 $34,000 of the amount over

Storage Tanks. The volume of the gasoline storage tank, in cubic feet, is given by the polynomial function V(r) 4.2r 3 37.7r 2 V, where is the radius in feet of the cylindrical part of the tank. What is the capacity of the tank if its radius is 4 feet? r 12 ft

Roller Coasters. The polynomial function (x) 0.001x 3  0.12x 2 3.6x 10 67 models the path of a portion of the track of a roller coaster. Use the function equation to find the height of the track for x 0, 20, 40, and 60. y f(x) = 0.001x3 0.12x2 + 3.6x + 10 10 20 30 40 Meters Meters 50 60 70

Customer Service. A software service hotline has found that on Mondays, the polynomial function C(t) 0.0625t 16t 4 x approximates the number of callers to the hotline at any one time. Here, represents the time, in hours, since the hotline opened at 8:00 A.M. How many service technicians should be on duty on Mondays at noon if the company doesnt want any callers to the hotline waiting to be helped by a technician?

Rain Gutters. A rectangular sheet of metal will be used to make a rain gutter by bending up its sides, as shown. If the ends are covered, the capacity of the gutter is a polynomial function of (x) 240x 2 x 1,440x: . Find the capacity of the gutter if is 3 inches. 120 in. 12 in.

Stopping Distances. The number of feet that a car travels before stopping depends on the drivers reaction time and the braking distance. For one driver, the stopping distance , in feet, is given by the polynomial function d(v)  0.04v v 2  0.9v , where is the velocity of the car in mph. Find the stopping distance at 60 mph.

Explain why 8 is not in the domain of the function f(x)  5x 7 x 8 d.

Explain why we can think of a function as a machine.

Consider the function defined by . Why do you think is called the independent variable and the dependent variable?

A website selling nutritional supplements contains the following sentence: Health is a function of proper nutrition. Explain what this statement means.

Solve each equation. If the equation is an identity or a contradiction, so indicate. 2(t  4)  5t  1  3(t 4)  7 6

Solve each equation. If the equation is an identity or a contradiction, so indicate. 3 2 (a 4)  2(a 3) a 2 2(

Let , function g be defined by {(4, 6), (6, 8), (8, 4)}, and function h be defined by the arrow diagram below. Find (8)  g(8) h(8) . 4 6 8 8 4 6 x

Find the domain of (x)  . 1 5[9(x 2) 6(x 3)  3] 4

Functions whose graphs are not lines are called _____ functions.

The graph of (x)  x 2 is a cuplike shape called a ______ .

The set of ______ real numbers is the set of real numbers greater than or equal to 0.

A shift of the graph of a function upward or downward is called a vertical _____.

Graph each basic function by plotting points and give its name. a. (x)  x 2 b. (x)  x 3 c. (x)  0 x 0

Complete each sentence about finding function values graphically. a. To find (3) y, we find the -coordinate of the point on the graph whose -coordinate is . b. To find the value of for which (x)  2 , we find the -coordinate of the point(s) on the graph whose -coordinate is . c. Suppose for a function that (5)  9. The corresponding ordered pair that will be on the graph of the function is .

Fill in the blank. The graph of g(x)  x2 is the ______ of the graph of (x)  x about the _____ x -axis.

Fill in the blanks. The illustration shows the projection of the graph of function on the . We see that the of is the set of real numbers less than or equal to 0. y f 11 1 2 3 234 2 3 4

Consider the graph of the function . a. Label each arrow in the illustration with the appropriate term: domain or range. b. Give the domain and range of . 111 1 2 3 4 5 2 3 3 2 2345

The graph of (x)  x 2  k for three values of is shown on the right. Find the value of k for a. the blue graph b. the red graph c. the green graph 6 2 3 4 4 23 234

The graph of (x)  0 x  h 0 for three values of is shown. Find the value of h for a. the blue graph b. the red graph c. the green graph y x 111 1 2 3 4 8 234567 234 56

a. Translate each point plotted b. Translate each point plotted on the graph below to the on the graph below to the left 5 units and then up right 4 units and then down 1 unit. 3 units. y x 111 2 3 1 2 3 4 5 23456 2 y x 111 2 3 4 5 1 2 3 2 2 3 4 5 6

a. Give the coordinates of the points where the given vertical line intersects the graph in red. b. Is this the graph of a function? Explain. y x 111 2 1 2 3 4 5 3 4 5 234

Fill in the blanks. Fill in the blanks. The line test: If a vertical line intersects a graph in more than one point, the graph is not the graph of a _____ .

Fill in the blanks. a. The graph of (x)  (x  4)3 is the same as the graph of (x)  x 3 except that it is shifted ___ units to the ___ . b. The graph of (x)  x 3  4 is the same as the graph of (x)  x 3 except that it is shifted ____units ____ .

Fill in the blanks. a. The graph of (x)  0 x 0 5 is the same as the (x)  0 x 0 graph of except that it is shifted ____ units ____ . b. The graph of (x)  0 x 5 0 is the same as the graph of (x)  0 x 0 except that it is shifted ____ units to the ___ .

Refer to the given graph to find each value. See Example 1. a .(2) b. (0) c. The value of for which (x)  4 d. The value of for which (x)  2 y f 111 2 3 4 1 2 3 4 234 2 3 4

Refer to the given graph to find each value. See Example 1. a. s(3) b. s(3) c. The values of for which s(x)  0 d. The values of for which s(x)  3 y 111 2 3 4 5 1 2 3 4 5 2345 2 3 4 5

Refer to the given graph to find each value. See Example 1. a. g(2) b. g(0) c. The value of for which g(x)  3 d. The values of for which x g(x)  1 g 111 2 3 1 2 3 234 2

Refer to the given graph to find each value. See Example 1. a. h(3) b. h(4) c. The values of for which h(x)  1 d. The value of for which h(x)  0 y x h 111 1 2 3 4 23

Find the domain and range of each function. See Objective 2 and Example 2. y f 111 2 3 4 1 2 3 4 234 2 3 4

Find the domain and range of each function. See Objective 2 and Example 2. y f 111 2 3 4 1 2 3 4 234 2 3 4

Find the domain and range of each function. See Objective 2 and Example 2. f 111 1 2 3 4 5 6 7 23 2 3 4 5

Find the domain and range of each function. See Objective 2 and Example 2. y f 111 1 2 3 4 5 6 7 23 2 3 4 5

Find the domain and range of each function. See Objective 2 and Example 2. y x 111 1 2 3 4 2 3 4 2345 2 3

Find the domain and range of each function. See Objective 2 and Example 2. y x 111 1 2 2 3 4 5 6 234 2 3 4

Find the domain and range of each function. See Objective 2 and Example 2. y f 111 2 3 4 1 2 3 4 234 2 3 4

Find the domain and range of each function. See Objective 2 and Example 2. y f 111 2 3 4 1 2 3 4 234 2 3 4

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. (x)  x 4 2  2

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. (x)  x 2 (x)  x 4 2

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. (x)  x  2 3 3

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. (x)  x 3 (x)  x  2

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. (x)  0 x 1 0 (

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. (x)  0 x  4

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. (x)  (x  4)2

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. (x)  (x 1)3

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  0 x 0 2

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  0 x  2 0

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  (x  1)3

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  x 3  5

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  x 2 3

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  (x 6)2 g

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  (x 4)3

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  0 x 0  1

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  x 3  4

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  x 2 5

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  (x  4)2

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using a translation and give its domain and range. See Examples 5 and 6. g(x)  (x 1)3

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  0 x 2 0 1 1

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  (x  2)2

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  (x  1) g(x)  0 x  4 0  3 3 2

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  0 x  4 0  3

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  (x 2)  3 2  4

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  (x 4)2 g(x)  (x 2)  3 2

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  0 x  3 0  5

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  (x 3)2 g(x)  0 x  3 0  5 2 g

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  x g(x)  0 x 0 3 g

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  0 x 0 3

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  x 2 g

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  (x  1)2

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  0 x  5 0

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  (x  4)3 g

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  x g(x)  0 x 0 4 2  3 g(

For each of the following functions, first sketch the graph of its associated function, , , or . Then draw the graph of function g using translations and/or a reflection. See Examples 7 and 8. g(x)  0 x 0 4 2

Use the graph of the function to find each of the following. See Example 9. Find: a. (1) b. (3) c. The values of for which (x)  0 d. The domain and range of f y x 11 5 5 10 15 10 15 234 2

Use the graph of the function to find each of the following. See Example 9. Find: a. (1) b. (0) c. The values of for which (x)  3 d. The domain and range of f 11 1 1 2 3 2 3 23 2 3

Use the graph of the function to find each of the following. See Example 9. Find: a. g(1) b. g(4) c. The values of x for which g(x)  4 d. The domain and range of g y g 11 1 1 2 3 4 2 3 4 5 23

Use the graph of the function to find each of the following. See Example 9. Find: a. h(0.5) b. h(0.5) c. The values of x for which h(x)  1.5 d. The domain and range of h y h 11 1 1 2 3 2 3 23 2 3

Determine whether each graph is the graph of a function. If it is not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 10. y 111 2 1 2 3 3 23

Determine whether each graph is the graph of a function. If it is not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 10. y 11 1 2 3 4 5 6 23 2 3

Determine whether each graph is the graph of a function. If it is not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 10. y 111 2 1 2 3 3 23 2 3

Determine whether each graph is the graph of a function. If it is not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 10. y 111 2 1 2 3 3 23 2 3

Determine whether each graph is the graph of a function. If it is not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 10. y x 111 2 1 2 3 4 3 4 234 2 3 4

Determine whether each graph is the graph of a function. If it is not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 10. y x 111 2 1 3 4 5 6 7 2 2 3 4 5 6

Determine whether each graph is the graph of a function. If it is not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 10. y x 111 2 1 2 3 4 3 4 23 2 3 4 5

Determine whether each graph is the graph of a function. If it is not, find two ordered pairs where more than one value of y corresponds to a single value of x. See Example 10. y x 111 2 1 2 3 4 3 4 234 2 3 4

Graph each function using window settings of [4, 4] for x and [4, 4] for y. The graph is not what it appears to be. Pick a better viewing window and find a better representation of the true graph. See Using Your Calculator: Graphing Functions. (x)  x 8 2  8

Graph each function using window settings of [4, 4] for x and [4, 4] for y. The graph is not what it appears to be. Pick a better viewing window and find a better representation of the true graph. See Using Your Calculator: Graphing Functions. (x)  x 3 8 2

Graph each function using window settings of [4, 4] for x and [4, 4] for y. The graph is not what it appears to be. Pick a better viewing window and find a better representation of the true graph. See Using Your Calculator: Graphing Functions. (x)  0 x  5 0

Graph each function using window settings of [4, 4] for x and [4, 4] for y. The graph is not what it appears to be. Pick a better viewing window and find a better representation of the true graph. See Using Your Calculator: Graphing Functions. (x)  0 x 5 0

Graph each function using window settings of [4, 4] for x and [4, 4] for y. The graph is not what it appears to be. Pick a better viewing window and find a better representation of the true graph. See Using Your Calculator: Graphing Functions. (x)  (x 6)2

Graph each function using window settings of [4, 4] for x and [4, 4] for y. The graph is not what it appears to be. Pick a better viewing window and find a better representation of the true graph. See Using Your Calculator: Graphing Functions. (x)  (x  9)2

Graph each function using window settings of [4, 4] for x and [4, 4] for y. The graph is not what it appears to be. Pick a better viewing window and find a better representation of the true graph. See Using Your Calculator: Graphing Functions. (x)  x 3  8

Graph each function using window settings of [4, 4] for x and [4, 4] for y. The graph is not what it appears to be. Pick a better viewing window and find a better representation of the true graph. See Using Your Calculator: Graphing Functions. (x)  x 3 12

Optics. See the illustration. The law of reflection states that the angle of reflection is equal to the angle of incidence. What function studied in this section models the path of the reflected light beam with an angle of incidence measuring ? Angle of reflection Reflected beam Angle of

Billiards. In the illustration, a rectangular coordinate system has been superimposed over a billiard table. Write a function that models the path of the ball that is shown banking off of the cushion. y 2 2 2 6 4 2 4 6

Center of Gravity. See the illustration. As a diver performs a 11 2-somersault in the tuck position, her center of gravity follows a path that can be described by a graph shape studied in this section. What graph shape is that?

Earths Atmosphere. The illustration below shows a graph of the temperatures of the atmosphere at various altitudes above Earths surface. The temperature is expressed in degrees Kelvin, a scale widely used in scientific work. a. Estimate the coordinates of three points on the graph that have an -coordinate of 200. b. Explain why this is not the graph of a function. 100 50 40 30 20 10 90 80 70 60 100 110 120 0 Temperature (Kelvin) Height (km) 200 300 400 x y Ionosphere Thermosphere Mesosphere Ozone layer Stratosphere Troposphere

Labor Statistics. The polynomial function that is graphed below approximates the number of manufacturing jobs (in millions) in the United States, where x is the number of years after 2000. Use the graph to answer the following questions. a. Estimate . Explain what the result means. b. Estimate the value of x for which . Explain what the result means. Years after 2000 Manufacturing employment (millions of jobs) 0 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 J

Transportation Engineering. The polynomial function A that is graphed below approximates the number of accidents per mile in one year on a 4-lane interstate, where is the average daily traffic in number of vehicles. Use the graph to answer the following questions. a. Estimate . Explain what the result means. b. Estimate the value of for which . Explain what the result means. A(x) x Average daily traffic (in number of vehicles) Number of accidents per mile in one year Source: Highway Safety Manual, Colorado Department of Transportation 10,000 20,000 30,000 A(x) = 0.000000000002x3 + 0.00000008x2 0.0006x + 2.45

Explain how to graph a function by plotting points.

Explain how to project the graph of a function onto the -axis. Give an example.

a. What does it mean to translate a graph vertically? b. What does it mean to horizontally translate a graph? c. What does it mean to reflect the graph of a function about the -axis?

A student was asked to determine whether the graph shown below is the graph of a function. What is wrong with the following reasoning? When I draw a vertical line through the graph, it intersects the graph only once. By the vertical line test, this is the graph of a function. y x 111 2 1 2 3 4 3 4 234 2 3 4

Explain why the graph of g(x)  (x 2)2 is two units to the right of the graph of f(x)  x2.

Explain why the range of the polynomial function graphed below is not (, ). y x 22 2 2 6 10 4 8 6 4 10 6 4 4 8 6 10

Solve each formula for the indicated variable. T W  ma for W

Solve each formula for the indicated variable. a  (n 1)d  l for n

Solve each formula for the indicated variable. s 1 2 gt2  vt for g

Solve each formula for the indicated variable. e  mc2 for m

Graph each function. (x)  e 0 x 0 for x 0 x 3 for x  0

Graph each function. (x)  e x 2 for x 0 0 x 0 for x  0

Find the domain and range of each function. a. 111 2 3 4 1 2 3 4 234 2 3 4 b. y 111 2 3 4 1 2 3 4 234 2 3 4 x

Light. Light beams coming from a bulb are reflected outward by a parabolic mirror as parallel rays. a. The cross-section of a parabolic mirror is given by the function for the following values of : , , , , , , , 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7. Sketch the parabolic mirror using the following grid. b. From the lightbulb filament at , draw a line segment representing a beam of light that strikes the mirror at (0.4, 0.16) and then reflects outward, parallel to the y-axis. y x 1.0 0.5 0.5 0.5 1.0 1.0

Complete the table and then graph the polynomial function. (x)  2x  6x 3 3x 2 11x  6x 0 1 2 3 4 1 2 3 x f(x

Complete the table and then graph the polynomial function. (x)  x 3 x 2  6x 3 0 1 2 3 1 2 3 4 x f(x)

The _____ of two sets is the set of elements that are common to both sets and the _____ of two sets is the set of elements that are in one set, or the other, or both.

x  3 and x  4 is a ______ inequality

6  x  1  1 is a linear ______ inequality.

(2, 8) is an example of an open ____ , [4, 0] is an example of a _____ interval, and (0, 9] is an example of a half- ____ interval.

a. The solution set of a compound inequality containing the word and includes all numbers that make _____ inequalities true. b. The solution set of a compound inequality containing the word or includes all numbers that make ____ , or the other, or _____ inequalities true.

The double inequality 4  3x  5  15 is equivalent to 4  3x  5 ______ 3x  5  15.

a. When solving a compound inequality containing the word and, the solution set is the ______ of the solution sets of the inequalities. b. When solving a compound inequality containing the word or, the solution set is the ____ of the solution sets of the inequalities.

When multiplying or dividing all three parts of a double inequality by a negative number, the direction of both inequality symbols must be _____ .

Use a check to determine whether -3 is a solution of the compound inequality. a. x 3  1  0 and 2x 3  10 x b. 2x  0 or 3x  5

Use a check to determine whether -3 is a solution of the double linear inequality. a. 1  3x  4  12 3 b. 1  3x  4  14

Use interval notation, if possible, to describe the intersection of each pair of graphs. a. b. c.

Use interval notation to describe the union of each pair of graphs. a. b. c.

Fill in the blanks: We read as ____ and as ____ .

Match each interval with its corresponding graph. a. [2, 3) b. (2, 3) c. [2, 3] i. [ ] 2 3 ii. [ ) 2 3 iii. ( ) 2 3

What set is represented by the interval notation (, )ii? Graph it.

a. Graph: (, 2) [3, ) b. Graph: ( (, 3) [2, )

Let A  {0, 1, 2, 3, 4, 5, 6}, B  {4, 6, 8, 10}, C  {3, 1, 0, 1, 2}, and D  {3, 1, 2, 5, 8}. Find each set. See Example 1. A B A

Let A  {0, 1, 2, 3, 4, 5, 6}, B  {4, 6, 8, 10}, C  {3, 1, 0, 1, 2}, and D  {3, 1, 2, 5, 8}. Find each set. See Example 1. A D

Let A  {0, 1, 2, 3, 4, 5, 6}, B  {4, 6, 8, 10}, C  {3, 1, 0, 1, 2}, and D  {3, 1, 2, 5, 8}. Find each set. See Example 1. C D B

Let A  {0, 1, 2, 3, 4, 5, 6}, B  {4, 6, 8, 10}, C  {3, 1, 0, 1, 2}, and D  {3, 1, 2, 5, 8}. Find each set. See Example 1. B C

Let A  {0, 1, 2, 3, 4, 5, 6}, B  {4, 6, 8, 10}, C  {3, 1, 0, 1, 2}, and D  {3, 1, 2, 5, 8}. Find each set. See Example 1. B C A

Let A  {0, 1, 2, 3, 4, 5, 6}, B  {4, 6, 8, 10}, C  {3, 1, 0, 1, 2}, and D  {3, 1, 2, 5, 8}. Find each set. See Example 1. A C

Let A  {0, 1, 2, 3, 4, 5, 6}, B  {4, 6, 8, 10}, C  {3, 1, 0, 1, 2}, and D  {3, 1, 2, 5, 8}. Find each set. See Example 1. A D C

Let A  {0, 1, 2, 3, 4, 5, 6}, B  {4, 6, 8, 10}, C  {3, 1, 0, 1, 2}, and D  {3, 1, 2, 5, 8}. Find each set. See Example 1. C D

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. See Examples 24. x  2 and x  5

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. See Examples 24. x  4 and x  7x

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. See Examples 24. 2x 1  3 and x  8  11

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. See Examples 24. 5x 3  2 and 6  4x 3

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. See Examples 24. 6x  1  5x 3  and x 2  9  6

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. See Examples 24. 2 3 x  1  9 and 3 4 x 1  10 2

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. See Examples 24. x  2  and 6x  9x 1 3 x 3

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. See Examples 24. 3 2 x  1 5  5 and 2x  1  9

Solve each double inequality. Graph the solution set and write it using interval notation. See Example 5. 4  x  3  7

Solve each double inequality. Graph the solution set and write it using interval notation. See Example 5. 5.3  x 2.3  1.3 2x

Solve each double inequality. Graph the solution set and write it using interval notation. See Example 5. 0.9  2x 0.7  1.5 7

Solve each double inequality. Graph the solution set and write it using interval notation. See Example 5. 7  3x 2  25

Solve each compound inequality. Graph the solution set and write it using interval notation. See Examples 6 and 7. x  2 or x  6

Solve each compound inequality. Graph the solution set and write it using interval notation. See Examples 6 and 7. x  1 or x  3x

Solve each compound inequality. Graph the solution set and write it using interval notation. See Examples 6 and 7. x 3  4 or x  2  0x

Solve each compound inequality. Graph the solution set and write it using interval notation. See Examples 6 and 7. 4x  12  or x 2  4

Solve each compound inequality. Graph the solution set and write it using interval notation. See Examples 6 and 7. 3x  2  8 or 2x 3  11

Solve each compound inequality. Graph the solution set and write it using interval notation. See Examples 6 and 7. 3x  4  2 or 3x  4  10

Solve each compound inequality. Graph the solution set and write it using interval notation. See Examples 6 and 7. 2x  x  3 or x 8  1  13 8

Solve each compound inequality. Graph the solution set and write it using interval notation. See Examples 6 and 7. 2(x  2)  x 11  or x 5  20

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 4(x  2)  12 or 3x  8  11

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 4.5x 1  10 or 6 2x  12 4

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 2.2x  19.8 and 4x  40

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 1 2x 2 and 0.75x  6

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 2  b 3  5 2

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 2  t 2  9 2

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 4.5x 2  2.5 or x  1 2x 1

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 0  x or 3x 5  4x 7

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 5(x 2)  0 and 3x  9

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. x 1  2(x  2) and x  2x 5

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. x  2x and 3x  2x

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. x 4  2.5 and 9x  2(4x  5) x

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 6  3(x 4)  24 9

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 4  2(x  8)  8 6

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 2x  1  5 and 3(x  1)  9 4

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 2(2)  3x 1 and 3x 1  1 3 2x 

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 4.5x 12 2  x or 15.3  3(x 1.4) 4.5

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. y  0.52  1.05y or 9.8 15y  15.7

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. x 0.7  5  4 and 4.8  3x 0.125 x

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. (x  1)  4(x  3) or x  12  3

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 24  3 2 x 6  15 5(x

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 4  2 3 x 2  6 24

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. x 3 x 4  1 6 or x 2  2 3  3 4

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. a 2  7 4  5 or 3 8  a 3  5 12

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 0  4 x 3  2

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 2  5 3x 2  2 0

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. x  6 x  1  2 x and x  1  3

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 3ax  2(x  2 3  7 and 2(x  2)  2

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 6  f(x)  0 f where f(x) 3x 9

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. 3  f(x)  7 where f(x) 2 3 x 1 3 3

Let f(x) 5x  14 g and g(x) 2x  8. Find all values of x for which f(x)  29 and g(x)  20 .

Let f(x) x 2 f. Find all values of x for which f(x)  5 or f(x)  1.

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the compound inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a.3x 2  4 x and 3x 2  4 x b. 3x 2  4 or x  6  12 and 3x 2  4 x  6  12

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the compound inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. x  2  10 or x 3  2 8 b. x  2  10 and x 3  2

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the compound inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. 2x  1  7 and 3x  5  23 b. 2x  1  7 or 3x  5  23

Solve the inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the compound inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. 7  4x  1  23 b. 7  4x  1  23

Baby Furniture. Refer to the illustration. A company manufactures various sizes of play yard cribs having perimeters between 128 and 192 inches, inclusive. a. Complete the double inequality that describes the range of the perimeters of the play yard shown.  4s  b. Solve the double inequality to find the range of the side lengths of the play yard. s s

Trucking. The distance that a truck can travel in 8 hours, at a constant rate of mph, is given by . A trucker wants to travel at least 350 miles, and company regulations dont allow him to exceed 450 miles in one 8-hour shift. a. Complete the double inequality that describes the mileage range of the truck.  8r  b. Solve the double inequality to find the range of the average rate (speed) of the truck for the 8-hour trip.

Thermostats. During business hours, as shown in figure (a), the Temp range control in an office is set at 5. This means that the heater comes on when the room temperature gets 5 degrees below the Temp setting and the air conditioner comes on when the room temperature gets 5 degrees above the Temp setting. a. Use interval notation to describe the temperature range when neither the heater nor the air conditioner will come on during business hours. b. After business hours, the Temp range setting is changed to save energy. See figure (b). Use interval notation to describe the after-business-hours temperature range when neither the heater nor the air conditioner will come on. Thermostat During business hours (a) 60 64 68 72 76 80 Temp range 10 . . . . 5 . . . . 0 Temp setting Thermostat After business hours (b) 60 64 68 72 76 80 Temp range

Treating Fevers. Use the flow chart to determine what action should be taken for a 13-month-old child who has had a temperature for 3 days and is not suffering any other symptoms. represents the childs temperature, the childs age in months, and the number of hours the child has experienced the symptoms. Is T 101 and A < 3? Is 3 A 12 and S > 24? Has fever shown no improvement in last 72 hours or is S > 120? Is there sore throat, ear pain, cough, abdominal pains, skin rash, diarrhea, urinary frequency, or other symptoms? Apply home treatment. Is there stiffness of the neck, confusion, marked irritability, rapid breathing, lethargy, or a seizure? See doctor now. Call doctor today. See doctor today. See the applicable article about the specific problem.

U.S. Health Care. Refer to the following graph. Let represent the percent of children covered by private insurance, the percent covered by Medicaid, and the percent not covered. For what years are the following true? a. P  63 or M  26 b. P  60 or M  29 c. M  29 and N  10 d. M  30 or N  9.2 U.S. Health Care Coverage for People Under 18 Years of Age (in percent) Private insurance Medicaid Not covered 2004 2005 2006 2007 63.2 62.1 59.4 59.8 26.4 27.2 29.9 29.8 9.2 9.3 9.5 9.0

Polls. For each response to the poll question shown below, the margin of error is (read as plus or minus) 2.8%. This means that for the statistical methods used to do the polling, the actual response could be as much as 2.8 points more or 2.8 points less than shown. Use interval notation to describe the possible interval (in percent) for each response. 2,258 adults were asked, Do you know the names of all your neighbors, most of them, some of them, or none of them? The responses were as follows. 19% 24% 29% 28% All of them Most of them Some of them None of them

Street Intersections. Refer to figure (a) below. a. Shade the area that represents the intersection of the two streets shown in the illustration. b. Shade the area that represents the union of the two streets. No Stopping Any Time No Stopping Any Time

Traffic Signs. The pair of signs shown in figure (b) above are a real-life example of which concept discussed in this section?

Explain how to find the union and how to find the intersection of (, 5) and (2, ) graphically

Explain why the double inequality 2  x  8 can be written in the equivalent form 2  x and x  8.

Explain the meaning of the notation for each type of graph. a. 2 1 0 1 2 b. y x 111 1 2 3 2 3 23 2 3

The meaning of the word or in a compound inequality differs from our everyday use of the word. Explain the difference.

Describe each set in words. a. (3, 3) b. [7, 12] c. (, 5] (6, )

What is incorrect about the double inequality 3  3x  4  3?

Airplanes. Together, a Delta B747 and a Delta B777 seat 681 passengers. If the B777 seats 125 less people than the B747, how many passengers does each seat? (Source: deltaskymag.com)

Denzel. As of October 2010, Denzel Washingtons three top domestic grossing films, American Gangster, Remember the Titans, and The Pelican Brief, had earned a total of $346.7 million. If American Gangster earned $14.5 million more than Remember the Titans, and if Remember the Titans earned $14.9 million more than The Pelican Brief, how much did each film earn as of that date? (Source: boxofficemojo.com)

Solve each compound inequality. Graph the solution set and write it in interval notation. 5 x  2 2 or 2x  10  30

Solve each compound inequality. Graph the solution set and write it in interval notation. 2   3 x 4 3  0 5 and x 5 2   3 x

Solve each compound inequality. Graph the solution set and write it in interval notation. x 12  4x  2x  16

Solve each compound inequality. Graph the solution set and write it in interval notation. 6(x 3)  3(3x  2)  4(2x  3)

The ____ ____ of a number is its distance from 0 on a number line.

2x 1 0 is an absolute value _____ and 2x 1 0 is an absolute value _____.

To _____ the absolute value in 0 3 x 0 4 50, we add 4 to both sides.

When we say that the absolute value equation and a compound equation are equivalent, we mean that they have the same _____.

When two equations are joined by the word or, such as x  1 5 x or x  1 5, we call the statement a _____ equation.

(x) 0 6x 2 0 is called an absolute value _____.

To solve absolute value equations and inequalities, we write and solve equivalent _____ equations and inequalities.

Two absolute value expressions are equal when the expressions within the absolute value bars are equal to or ______ of each other.

Consider the following real numbers: 3, 2.01, 2, 1.99, 1, 0, 1, 1.99, 2, 2.01, 3 a. Which of them make 0 x 0 2 true? b. Which of them make x 0  2 true? c. Which of them make x 0  2 true?

Determine whether is a solution of the given equation or inequality. a. x 1 0 4 0 b. x 1 0  4 c. x 1 0  4 0 d. 5 x 0 0 x  12 0

For each absolute value equation, write an equivalent compound equation. a. x 7 0 8 0 is equivalent to x 7 x or x 7 0 b. x  10 0 0 x 3 0x is equivalent to x  10 x or x  10

For each absolute value inequality, write an equivalent compound inequality. a. x  5 0  1 is equivalent to  x  5  b. x 6 0  3 is equivalent to x 6  or x 6 

For each absolute value equation or inequality, write an equivalent compound equation or inequality. a. 0 x 0 8 0 b. 0 x 0  8 c. x 0  8 d. 0 5x 1 0 0 x  3 0

Perform the necessary steps to isolate the absolute value expression on one side of the equation. Do not solve. a. 0 3x  2 0 7 5 b. 6  2 0 5x 19 0  40

Determine the solution set of each absolute value equation or inequality by inspection. (No work is necessary.) Your answer should be either all real numbers or no solution. a. 0 7x  6 0 8 0 b. 7x  6 0  8 c. 7x  6 0  8

Write the inequality 10  0 16x 3 in an equivalent form with the absolute value expression on the left side.

Match each equation or inequality with its graph. a. x 0 1 b. 0 x 0  1 c. x 0  1 i. ) 1 1 ii. 1 0 1 iii. ) ( 1 1

Describe the set graphed below using interval notation.

Solve each equation. See Example 1. 0 x 0 23 0

Solve each equation. See Example 1. x 0 90

Solve each equation. See Example 1. x 5 0 8 0

Solve each equation. See Example 1. x 7 0 4 0

Solve each equation. See Example 1. 3x  2 0 16 0

Solve each equation. See Example 1. 5x 3 0 22 0

Solve each equation. See Example 1. x 5 ` 10

Solve each equation. See Example 1. x 7 ` ` 2 x

Solve each equation. See Example 1. 2x  3.6 0 9.8 0

Solve each equation. See Example 1. 4x 24.8 0 32.4

Solve each equation. See Example 1. 7 2 x  3 ` 5 0

Solve each equation. See Example 1. x 2.1 0 16.3 7

Solve each equation. See Example 2. x 3 0 19 3 0 x

Solve each equation. See Example 2. 0 x 10 0  30 50

Solve each equation. See Example 2. 3x 7 0  8 22 0

Solve each equation. See Example 2. 6x 3 0  7 28

Solve each equation. See Example 2. 0 3 4x 0  1 6 0

Solve each equation. See Example 2. 8 5x 0 8 10

Solve each equation. See Example 2. 7 8 x  5 ` 2 7 0

Solve each equation. See Example 2. 3 4 ` x  4 ` 5 11 7

Solve each equation. See Example 3. 5 x  2 ` 8 8

Solve each equation. See Example 3. 1 9 ` x  4 `  25 25 1

Solve each equation. See Example 3. 2 0 3x  24 0 0 

Solve each equation. See Example 3. 8 ` 2x 3 10 ` 0

Solve each equation. See Example 3. 5 0 2x 9 0  14 14 10

Solve each equation. See Example 3. 10 0 16x  4 0 3 3 8 `

Solve each equation. See Example 3. 6 3 0 10x  5 0 6 15

Solve each equation. See Example 3. 15 0 12x  12 0 15 5

Solve each equation. See Example 5. 0 5x 12 0 0 4x 16 0 0 4x

Solve each equation. See Example 5. 0 4x 7 0 0 3x 21 0 6

Solve each equation. See Example 5. 0 10x 0 0 x 18 0

Solve each equation. See Example 5. 0 6x 0 0 x  45 0 0

Solve each equation. See Example 5. 2 x 0 0 3x  2 0

Solve each equation. See Example 5. 0 4x  3 0 0 9 2x 0

Solve each equation. See Example 5. 5x 7 0 0 4(x  1) 0

Solve each equation. See Example 5. 2x  1 0 0 3(x  1) 0

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 6 and 8. 0 x 0  4

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 6 and 8. x 0  9

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 6 and 8. x  9 0  12

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 6 and 8. 0 x 8 0  12

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 6 and 8. 3x 2 0  10 0

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 6 and 8. 4 3x 0  13

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 6 and 8. 5x 12 0  5 0 3

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 6 and 8. 3x  2 0  3

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 911. x 0  3

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 911. x 0  7

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 911. 0 x 12 0  24 0

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 911. 0 x  5 0  7

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 911. 0  0 5x 1 0 2 0

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 911. 0  0 6x 3 0 5 0

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 911. 0 4x  3 0  5

Solve each inequality. Graph the solution set and write it using interval notation. See Examples 911. 7x  2 0  8 0

See Examples 4, 6, and 9. Let (x)  x 3 . For what value(s) of x is f (x) 3?

See Examples 4, 6, and 9. Let g(x) 2 x . For what value(s) of x is g(x) 2?

See Examples 4, 6, and 9. Let (x) 0 2(x 1)  4 0 x. For what value(s) of x is (x)  4?

See Examples 4, 6, and 9. Let h(x) 0 x x 5 1 2 0. For what value(s) of x is h(x)  9 10?

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 3x  2 0  1  15

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 2x 5 0 5  20 h

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 6 ` `  x 2 3 `  24

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 8 ` x 2 3 ` `  32 x

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 7 2 0 0.3x 3 0 1

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 1 1 0 0.1x  8 0 8 `

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 2 3x 0  8 0

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 1 2x 0  5

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 7x  12 0 0 x 6

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 8 x 0 0 x  2

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 2  3 0 2 3x 0  2

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 7  0 15x 45 0  7

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 14 0 x 3

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 75 0 x  4

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 6 5 ` 3x 5  x 2 `

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 11 12 ` x 3 3x 4 `

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 0 2x 3 0  7 0

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 0 3x  1 0  8

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 0 0.5x  1 0  23

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. 15  7 0 1.4x  9

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. a. x 10 1 1 0 b. x 10 1 ` 1 x c. x 10 1  1 d. x 10 1 `  1 x

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. a. 4x 5 15 0 b. 4x 5 0 15 c. 4x 5  15 0 d. 4x 5 0  15

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. a. 0.9 0.3x 8.4 0 b. 0.9 0.3x 0 8.4 c. 0.9 0.3x  8.4 0 d. 0 0.9 0.3x 0  8.4

Solve each equation and inequality. For the inequalities, graph the solution set and write it using interval notation. a. 8(x 4) 6x 44 b. 8(x 4) 0 0 6x 44 c.

Solve the absolute value inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the absolute value inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. 8x 40 0  16 0 b. 0 8x 40 0  16

Solve the absolute value inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the absolute value inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. 0  0 14 27x 0 b. 0  0 14 27x 0

Solve the absolute value inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the absolute value inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. 4x 4 3 ` 1  11 0 b. 4x 4 3 ` 1  11

Solve the absolute value inequality in part a. Graph the solution set and write it in interval notation. Then use your work from part a to determine the solution set for the absolute value inequality in part b. (No new work is necessary!) Graph the solution set and write it in interval notation. a. 1 2 x 3 `  2  7 b. ` 1 2 x 3 `  2  7

Temperature Ranges. The temperatures on a sunny summer day satisfied the inequality t 780  8 t, where is a temperature in degrees Fahrenheit. Solve this inequality and express the range of temperatures as a double inequality.

Operating Temperatures. A car CD player has an operating temperature of 0 t 400  80 t, where is a temperature in degrees Fahrenheit. Solve the inequality and express this range of temperatures as an interval.

Auto Mechanics. On most cars, the bottoms of the front wheels are closer together than the tops, creating a camber angle. This lessens road shock to the steering system. (See the illustration.) The specifications for a certain car state that the camber angle of its wheels should be . a. Express the range with an inequality containing absolute value symbols. b. Solve the inequality and express this range of camber angles as an interval. Camber angle Axle

Steel Production. A sheet of steel is to be 0.250 inch thick with a tolerance of 0.025 inch. a. Express this specification with an inequality containing absolute value symbols, using to represent the thickness of a sheet of steel. b. Solve the inequality and express the range of thickness as an interval.

Error Analysis. In a lab, students measured the percent of copper in a sample of copper sulfate. The students know that copper sulfate is actually 25.46% copper by mass. They are to compare their results to the actual value and find the amount of experimental error. Which measurements shown in the illustration satisfy the absolute value inequality p 25.46 0  1.00? Lab 4 Title: "Percent copper (Cu) in copper sulfate (CuSO4 5H2O)" Section A Results Trial #1: 22.91% Trial #2: 26.45% Trial #3: 26.49% Trial #4: 24.76%

Error Analysis. See Exercise 107. Which measurements satisfy the absolute value inequality p 25.46 0  1.00?

Explain the error. Solve: 0 x 0  2 6 x  2 6 or x  2 6 x 4 x 8 x

Explain why the equation 0 x 4 0 5x has no solution.

Explain the differences between the solution sets of x 0 8, x 0  8 and x 0  8.

Explain how to use the graph in the illustration to solve the following. a. 0 x 2 0 3 b. 0 x 2 0  3 c. x 2 0  3 y 111 1 2 3 4 5 2 3 2 2 3

Flutes. When it is assembled, a flute is 29 inches long. The middle piece is 4 inches less than twice as long as the first piece. The last piece is two-thirds as long as the first piece. Find the length of each piece of the flute. First piece Middle piece Last piece

Commercials. For the typical one-hour prime-time television slot, the number of minutes of commercials is 3 7 of the number of minutes of the actual program. Determine how many minutes of the program are shown in that one hour.

a. For what values of k does x 0  k 0 have exactly two solutions? b. For what values of k does x 0  k 0 have exactly one solution?

Solve: 20 2x3 0 64 k

When we write 2x  4 as 2(x  2), we say that we have ______ x  4.2

When we factor a polynomial, we write a sum of terms as a _____ of factors.

The abbreviation GCF stands for _____ _____ _____.

If a polynomial cannot be factored, it is called a ______ polymonial or an irreducible polymonial.

To factor ab  6a  2b  12 by ______, we begin by factoring out from the first two terms and 2 from the last two terms.

The trinomial 4a2 5a 6 is written in ______ powers of a.

The _____ coefficient of x2 3x  2 is 1, the _____ of the middle term is -3, and the last term is .

The statement x2 x 12 (x 4)(x  3) shows that x2 x 12x factors into the _____ of two binomials.

The prime factorizations of three terms are shown here. Find their GCF. 2  2  3  x  x  y  y  y 2  3  3  x  y  y  y  y 2  3  3  7  x  x  x  y  y

Use multiplication to determine whether (3t 1)(5t 6) is the correct factorization of 15t 2 19t  6 .

Complete the table. Sum of the Factors of 8 factors of 8 2(4) 8 1(8) 8 2(4) 8 1(8) 8

Find two integers whose a. product is 10 and whose sum is 7. b. product is 8 and whose sum is -6. c. product is -6 and whose sum is 1. d. product is -9 and whose sum is -8.

Complete the key number table. Key number 12 b 7 Negative factors of 12 Sum of the factors of 12 3(4) 12 1(12) 12 14. U

Use the substitution x = a b to rewrite the trinomial 6 (a b)2 17 (a b)3.

Complete each factorization. 15c3 d4 25 c2 d2 5 c3 d6 (3c  5  cd2)

Complete each factorization. x3 x2  2x 2 (x 1)  (x 1) 15c ( )(x2  2)

Complete each factorization. 6m2  7m 3 ( 1)(2m  )

Complete each factorization. 2y 2  10y  12 (y 2  5y  6) 2(y  )(  2)

Factor each polynomial. See Example 1. 2x 2 6x

Factor each polynomial. See Example 1. 3y 3  3y 2

Factor each polynomial. See Example 1. 15x2 y 10x 2 y2

Factor each polynomial. See Example 1. 63x 3 y2  81x 2 y4 1

Factor each polynomial. See Example 1. 27z 3  12z 2  3z

Factor each polynomial. See Example 1. 25t 6 10t 3  5t 2 27

Factor each polynomial. See Example 1. 24s 3 12s 2 t  6st2

Factor each polynomial. See Example 1. 18y2 z 2  12y 2 z 3 24y 4 z 3 24

Factor each polynomial. See Example 1. 11x3 12y

Factor each polynomial. See Example 1. 14s 3  15t 6 1

Factor each polynomial. See Example 1. 23a2 b3  4x 3 y 2

Factor each polynomial. See Example 1. 18p3 q2 5t 5 2

Factor each polynomial by factoring out the opposite of the GCF. See Example 2. 8a 16 6b

Factor each polynomial by factoring out the opposite of the GCF. See Example 2. 6b 30 1

Factor each polynomial by factoring out the opposite of the GCF. See Example 2. 6x 2 3xy 8

Factor each polynomial by factoring out the opposite of the GCF. See Example 2. 15y 3 25y 2 6x

Factor each polynomial by factoring out the opposite of the GCF. See Example 2. 18a2 b  12ab2 1

Factor each polynomial by factoring out the opposite of the GCF. See Example 2. 21t 5  28t 3 18

Factor each polynomial by factoring out the opposite of the GCF. See Example 2. 8a4 c8  28a3 c8 20a2 c 9 2

Factor each polynomial by factoring out the opposite of the GCF. See Example 2. 30x10y  24x9 y 2 60x8 y 2 8a

Factor. See Example 3. (x  y)u  (x  y)v

Factor. See Example 3. 4(x  y)  t(x  y)

Factor. See Example 3. 5(a b  c) t(a b  c)

Factor. See Example 3. (a b c)r (a b c)s

Factor by grouping. See Example 4. ax  bx  ay  by

Factor by grouping. See Example 4. ar br  as bs

Factor by grouping. See Example 4. x  2  yx x y

Factor by grouping. See Example 4. d2  cd  c  d

Factor by grouping. See Example 4. t  3 3t 2 7t  21

Factor by grouping. See Example 4. b3 4b2 t 3b  12 3

Factor by grouping. See Example 4. a2 4b  ab 4a

Factor by grouping. See Example 4. 3c cd  3d c2

Factor. See Example 5. 6x3 6x2  12x 12

Factor. See Example 5. 3x 3 6x 2  15x 30 6

Factor. See Example 5. 28a3 b3 c  14a3 c 4b3 c 2c 3

Factor. See Example 5. 12x 3 z  12xy2 z 8x 2 yz 8y 3 z 2

Solve for the specified variable or expression. See Example 6. 2g ch  dh for h

Solve for the specified variable or expression. See Example 6. d1d2 d2  d1 for

Solve for the specified variable or expression. See Example 6. r1r2 rr2  rr1 for r1

Solve for the specified variable or expression. See Example 6. rx ty by for y

Solve for the specified variable or expression. See Example 6. b2 x 2  a2 y 2 a2 b2 for a2

Solve for the specified variable or expression. See Example 6. b2 x 2  a2 y 2 a2 b2 for b2

Solve for the specified variable or expression. See Example 6. Sn (n 2)180 for n

Solve for the specified variable or expression. See Example 6. S(1 r) a lr for r

Factor. See Example 7 or 12. x  2 5x  6

Factor. See Example 7 or 12. y 2  7y  6

Factor. See Example 7 or 12. x  2  x 30

Factor. See Example 7 or 12. c 2  3c 28 2

Factor. See Example 8. 3x 2  12xy 63y2

Factor. See Example 8. 2y2  4yz 48z 2 3x

Factor. See Example 8. 6a2 30ab  24b2 2

Factor. See Example 8. 4b2  12bc 16c 2 6

Factor. See Example 8. n4 28n3 t 60n2 t 2 4b

Factor. See Example 8. c 4 16c3 d 80c 2 d2

Factor. See Example 8. 3x 2  15xy 18y2

Factor. See Example 8. 2y 2 16yt  40t 2

Factor. See Example 9 or 12. 5x  2  13x  6

Factor. See Example 9 or 12. 5x2  18x  9

Factor. See Example 9 or 12. 7a  2  12a  5

Factor. See Example 9 or 12. 7a2 36a  5

Factor. See Example 9 or 12. 11y 2  32y 3 7a

Factor. See Example 9 or 12. 2y2 9y 18 2

Factor. See Example 9 or 12. 8x 2 22x  5 2y

Factor. See Example 9 or 12. 4z 2 13z  3 2

Factor. See Example 9 or 12. 6y 2 13y  6 4z

Factor. See Example 9 or 12. 6x 2 11x  3 2

Factor. See Example 9 or 12. 15b  2  4b 4 6

Factor. See Example 9 or 12. 8a2  6a 9 2

Factor each expression. See Example 10. 30x  4 25x 2 20 8

Factor each expression. See Example 10. 14x 4  77x 2  84

Factor each expression. See Example 10. 32x  4 96x 2  72

Factor each expression. See Example 10. 20a4  60a2  45

Factor each expression. See Example 10. 64h5 4h  24h3

Factor each expression. See Example 10. 9x 5 24x  30x 3 64

Factor each expression. See Example 10. 3a4 5a2 b2 2b4 9x

Factor each expression. See Example 10. 2x 4  3x 2 y 2  5y 4 3

Factor by using a substitution. See Example 11. (a  b) 2 2(a  b) 24

Factor by using a substitution. See Example 11. (x y) 2  3(x y) 10

Factor by using a substitution. See Example 11. (x  a) 2  2(x  a)  1

Factor by using a substitution. See Example 11. (a  b) 2 2(a  b)  1

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 3(m  n  p)  x(m  n  p)

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. x(x y z)  y(x y z)

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 63u3 v 6  28u2 v 7 21u3 v 3

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 56x 4 y 3 72x 3 y 4  80xy2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. b4 x 2 12b2 x 2  35x 2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. c 3 x 4  11c 3 x 2 42c 3

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. n m  mn

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. a2 x 2 10 2x 2  5a2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. x 2  4xy  21y 2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. a2 4ab  5b2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. a  2 x  bx a2 b

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. x 2 a y ax xy  a 2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 4y  2  4y  1

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 9x 2  6x  1 2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. b  2  8b  18

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. x 2  4x 28 2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 13r  3r 2 10

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. r  3r 2 r 10 2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. y 3 12  3y 4y 2 r

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. h3 8  h 8h2

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 2y  84y 5 26y 3  60y

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 2y 5 26y 3  84y 5

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 14(q r) 2 17(q r) 6 2y

Factor completely. Factor out all common factors first including -1 if the first term is negative. If an expression is prime, so indicate. 8(h  s) 2  34(h  s)  35

Crayons. The amount of colored wax used to make the crayon shown in the illustration can be found by computing its volume using the formula V pr 2 h1  1 3 pr 2 h2 Factor the expression on the right side of this equation. h2 h1 Crayon r

Packaging. The amount of cardboard needed to make the following cereal box can be found by finding the area , which is given by the formula A 2wh  4wl  2lh where is the width, the height, and the length. Solve the equation for the width. Lucky Snaps Delicious Light Crispy Lucky Snaps Delicious Light Crispy l l l l h w w w

Ice. The surface area of the ice cube is represented by the expression 6x2  36x  54. Use factoring to find the length of an edge of the cube.

Checkers. The area of the checkerboard is represented by the expression 25x2 40x  16. Use factoring to find the length of each side.

Explain the error in the following solution. Solve for r1: r1r2 rr2  rr1 r1r2 r2 rr2  rr1 r2 r1 rr2  rr1 r2

Explain the error. Factor: 2x 2 4x 6 (2x  2)(x 3) r1

Use the graph to find: a. s(5) b. s(4) c. The values of for which s(x) 0 d. The value of for which s(x) 4 y 111 2 3 4 5 1 2 3 4 5 2345 2 3 4 5

Determine whether each equation defines y to be a function of x. If it does not, find two ordered pairs where more than one value of y corresponds to a single value of x. a. x 0 y 0 b. y x 2 x c. y2 x

Factor out the specified factor. t3 from t 5  4t 6 t

Factor out the specified factor. 7x3n from 21x 6n  3n 7x  14

Factor. Assume that is a natural number. x 2 2n  2xn  1

Factor. Assume that is a natural number. 2a6n 3a3n x 2 2n

Factor. Assume that is a natural number. x 3 4n  2x2n y2n  y 4n

Factor. Assume that is a natural number. 6x 2n  7x n x 3 4

When the polynomial 4x2 25 is written as (2x)2 (5)2 4, we see that it is the difference of two _____ .

When the polynomial 8x3 25 is written as (2x)3 5(3), we see that it is the sum of two ____ .

a. Write the first ten perfect-square integers. b. Write the first ten perfect-cube integers.

a. Use multiplication to verify that the sum of two squares x2 25 does not factor as (x 5)(x 5) 2 . b. Use multiplication to verify that the difference of two squares x2 25 factors as (x 5)(x 5) 2 .

Complete each factorization. a. F2 L2  (F L)( ) b. F3 L3  (F L)( ) F c. F3 L3  (F L)( )F3

Factor each binomial. a. 5p2 20 b. 5p2 20 c. 5p3 20 d. 5p3 40

Give an example of each. a. a difference of two squares b. a square of a difference c. a sum of two squares d. a sum of two cubes e. a cube of a sum

Fill in the blanks. a. 36y 2 49m4  ( ) 2 ( ) 2 5 b. 125h3 27k 6  ( ) 3 ( ) 3 3

Factor. See Example 1. x2 16

Factor. See Example 1. y2 49

Factor. See Example 1. 9y2 64

Factor. See Example 1. 16x2 81

Factor. See Example 1. 144 c2

Factor. See Example 1. 25 t2

Factor. See Example 1. 100m 1 2 1

Factor. See Example 1. 144x 2 1

Factor. See Example 1. 81a2 49b2

Factor. See Example 1. 64r 2 121s 2 8

Factor. See Example 1. x 2 25 6

Factor. See Example 1. a2 36

Factor each difference of two squares. See Example 2. 9r 4 121s 2

Factor each difference of two squares. See Example 2. 81a4 16b2 9

Factor each difference of two squares. See Example 2. 16t 2 25w4

Factor each difference of two squares. See Example 2. 9r 2 25s 4 1

Factor each difference of two squares. See Example 2. 100r 2 s 4 t 4

Factor each difference of two squares. See Example 2. 400x 2 z 4 a4 1

Factor each difference of two squares. See Example 2. 36x 4 y 2 49z 6 4

Factor each difference of two squares. See Example 2. 4a2 b4 9d6

Factor completely. See Example 3. x4 y4

Factor completely. See Example 3. 16n4 1 x

Factor completely. See Example 3. 16a4 81b4

Factor completely. See Example 3. 81m4 256n4

Factor. See Example 4. (x y) 2 z 2 8

Factor. See Example 4. a2 (b c) 2 (x

Factor. See Example 4. (r s) 2 t 4 a

Factor. See Example 4. m n) 2 p4 (r

Factor each expression. Factor out any GCF first. See Example 5. 2x 2 288

Factor each expression. Factor out any GCF first. See Example 5. 8x2 72 2

Factor each expression. Factor out any GCF first. See Example 5. 3x3 243x

Factor each expression. Factor out any GCF first. See Example 5. 2x3 32x

Factor each expression. Factor out any GCF first. See Example 5. 5ab4 5a

Factor each expression. Factor out any GCF first. See Example 5. 3ac4 243a 4

Factor each expression. Factor out any GCF first. See Example 5. 64b 4b5

Factor each expression. Factor out any GCF first. See Example 5. ,250n 2n5 6

Factor by first grouping the appropriate terms. See Example 6. c2 d2 c d 1,25

Factor by first grouping the appropriate terms. See Example 6. s 2 t 2 s t 2 d

Factor by first grouping the appropriate terms. See Example 6. a 2 b2 2a 2b

Factor by first grouping the appropriate terms. See Example 6. m2 n2 a 3m 3n 2

Factor by first grouping the appropriate terms. See Example 6. x 2 12x 36 y2

Factor by first grouping the appropriate terms. See Example 6. x 2 6x 9 4y 2 x 2

Factor by first grouping the appropriate terms. See Example 6. x 2 2x 1 9z2

Factor by first grouping the appropriate terms. See Example 6. x 2 10x 25 16z 2 x 2

Factor each sum of cubes. See Example 7. a3 125

Factor each sum of cubes. See Example 7. b3 64

Factor each sum of cubes. See Example 7. 8r 3 s3

Factor each sum of cubes. See Example 7. 27t 3 u3

Factor each difference of cubes. See Example 8. 64t 6 27v3

Factor each difference of cubes. See Example 8. 125m3 x6

Factor each difference of cubes. See Example 8. x 3 216y6

Factor each difference of cubes. See Example 8. 8c 6 343w3

Factor. See Example 9. (a b) 3 27

Factor. See Example 9. (b c) 3 1,000

Factor. See Example 9. 64 (a b) 3

Factor. See Example 9. 1 (x y) 3 6

Factor each expression completely. Factor a difference of two squares first. See Example 10. x6 1

Factor each expression completely. Factor a difference of two squares first. See Example 10. x 6 y 6

Factor each expression completely. Factor a difference of two squares first. See Example 10. x 12 y 6

Factor each expression completely. Factor a difference of two squares first. See Example 10. a12 64

Factor each sum or difference of cubes. Factor out the GCF first. See Example 11. 5x 3 625

Factor each sum or difference of cubes. Factor out the GCF first. See Example 11. 2x3 128 3

Factor each sum or difference of cubes. Factor out the GCF first. See Example 11. 4x5 256x2

Factor each sum or difference of cubes. Factor out the GCF first. See Example 11. 2x6 54x3 4

Factor each expression. 64a3 125b6

Factor each expression. 8x 6 27y3

Factor each expression. 288b2 2b6

Factor each expression. 98x 2x5

Factor each expression. x 2 y2 8x 8y 98

Factor each expression. 5m 5n m2 n2

Factor each expression. x9 y9

Factor each expression. x6 y6

Factor each expression. 144a2 t 2 169b6 x

Factor each expression. 25x6 81y2 z 2

Factor each expression. 100a2 9b2

Factor each expression. 25s 4 16t 2 1

Factor each expression. 81c4 d4 16t 4

Factor each expression. 256x 4 81y4

Factor each expression. 128u2 v3 2t 3 u2 2

Factor each expression. 56rs2 t 3 7rs2 v6

Factor each expression. y2 (2x t) 2 56

Factor each expression. (15 r) 2 s 2 y

Factor each expression. x2 20x 100 9z 2 (1

Factor each expression. 49a2 b2 14b 49 x2

Factor each expression. (c d) 3 216 4

Factor each expression. 1 (x y) 3 (c

Factor each expression. 1 36 y 4 1

Factor each expression. 4 81 m4 1

Factor each expression. m6 64

Factor each expression. 1 y 6

Factor each expression. (a b)x 3 27(a b) 1

Factor each expression. (c d)r 3 (c d)s 3 (a

Factor each expression. x 9 y 12z 15

Factor each expression. r 12 s 18t 24

Factor each expression. a. q2 64 2 b. q3 64

Factor each expression. a. a2 b2 b. a3 b3 a

Factor each expression. a. d2 25 b. d3 125

Factor each expression. a. m2 27 b. m3 27

Factor the expression in part a. Then use your answer from part a to give the factorization of the expression in part b. (No new work is necessary!) a. a6 b3 b. a6 b3 a

Factor the expression in part a. Then use your answer from part a to give the factorization of the expression in part b. (No new work is necessary!) a. c3 1 8 b. c3 1 8

Factor the expression in part a. Then use your answer from part a to give the factorization of the expression in part b. (No new work is necessary!) a. 125m3 8n3 b. 125m3 8n3 1

Factor the expression in part a. Then use your answer from part a to give the factorization of the expression in part b. (No new work is necessary!) a. w3 0.001 b. w3 0.001 3

Candy. To find the amount of chocolate used in the outer coating of the malted-milk ball shown, we can find the volume of the chocolate shell using the formula Factor the expression on the right side of the formula. V  4 3 pr1 3 4 3 pr2 3 Outer radius r1 Inner radius r2

Movie Stunts. The function that gives the distance a stuntwoman is above the ground seconds after she falls over the side of a 144-foot tall building is h(t)  144 16t2. Factor the right side.

Explain how the patterns used to factor the sum and difference of two cubes are similar and how they differ.

Explain why the factorization is not complete. Factor: 1 t8 ( 1 t4)(1 t4)

Explain the error. Factor: 4g2 16 (2g 4) (2g 4)