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Date Created: 12/15/15
Chapter 1.1 Applications of Linear Equations A strategy for solving applied problems 1. Read the problem as many times as needed to understand it thoroughly. Pay close attention to the question asked to help identify the quantity a variable should represent. 2. Assign a variable to represent the quantity you are looking for and when necessary, express all other unknown quantities in terms of this variable. Frequently, it is helpful to draw a diagram to illustrate the problem or to set up a table to organize the information. 3. Write an equation that describes the situation. 4. Solve the equation. 5. Answer the question asked in the problem. 6. Check the answer against the description of the original problem (not just the equation solved in step 4.) Examples: 1. The width of a rectangle is 3 ft. more than onehalf its length, and the perimeter of the rectangle is 36 ft. Find its dimensions. 2. A retailer’s cost for a microwave oven is $480. If he wants to make a profit of 20% of the selling price, at what price should the oven be sold? 3. Angelina jogs 100 meters in the same time that Harry bicycles 150 meters. If Harry bicycles 15 meters per minute faster than Angelina jogs, find the rate of each. 4. A motorcycle police officer is chasing a car that is speeding at 70 miles per hour. The police officer is 1 mile behind the car and is traveling 80 miles per hour. How long will it be before the officer overtakes the car? Practice problems page 88 #68, 76, 88 1.2 Quadratic Equations Substitution Determine if the given value of a variable is a solution to the equation. o Is x = 9 a solution to the equation x – 8 x- 9 = 0 Solving quadratic equations by factoring Examples 2 x – 5x + 4 = 0 6x = 1 – x by the square root method—use when there is no ‘x’ term. Examples 2 2x = 50 by completing the square (Be sure you can do this method. We will use it again). Examples 2 x + 6x = -7 2 3k – 5k + 1 = 0 2 b b 4ac 2a 2 by using quadratic formula given ax + bx + c = 0 Examples 2 m + 3m + 2 = 0 2 x – 7 = 4x Practice Problems Page 101 #52, 54, 58, 65 1.4 Solving Other Types of Equations Solving a rational equation. find the LCD multiply both sides by the LCD set equal to 0 factor set each factor equal to 0 solve check for extraneous solutions. Example 2(x+1) x 9 x−2 − x+1 = 2 x −x−2 x 7 14x 2 x 7 x 7 x 49 Solving equations involving radicals isolate radical on one side of equation square (cube, etc) both sides and simplify if there is another radical, repeat steps 1 and 2 set each factor equal to 0 CHECK –Sometimes, clearing a radical produces a wrong solution. Examples 3 2x 3 3 m 1 m 5 Solve by substitution let u equal the x term factor set each factor equal to 0 replace u with x Examples x −7x +12=0 x 3 x 2 0 2 2 √+1 )−2 (2√t+1 )−3=0 1/2 1/4 2x – x – 1 = 0 Practice Problems page 126 #28, 32, 40, 44, 68 1.5 Solving Linear Inequalities using Test Points Rewrite the inequality so that the right side is 0. Factor the left side. Draw a number line. Find and plot the points where each factor is 0. Choose test points at each interval. Draw a sign chart. Graph the solution set. Give solution in interval notation. o Examples 2 4 x −2 x−2<0 2 4 x +12x≤−9 Solving Rational Inequalities using Test Points Rewrite the inequality so that the right side is 0. Simplify the fraction into one fraction—common denominator, simplify Factor the numerator. Set each factor in numerator and denominator to 0. Note: The factor(s) in the denominator tell us what the solution cannot be. Recall that a denominator of zero renders the fraction undefined. Place an open circle on these points. Choose test points at each interval. Draw a sign chart based on fraction and determine sign of solution. Graph the solution set. Give solution in interval notation. o Examples x >0 x−2 x−1 x−2 ≥3 x+3≤ x−1 x+1 x−2 Practice Problems page 140 #86, 112, 118 1.6 Solving Linear Inequalities Inequality symbols: > greater than > greater than or equal to < less than < less than or equal to We solve inequalities using the same methods to solve equations with one difference, which is shown in these two properties. 1. If a > b and c < 0, then ac < bc a b c c 2. If a > b and c < 0, then In other words, when you multiply or divide by a negative number you must flip the inequality sign. Examples: 2x + 3 > 4 -3x – 4 < 8 Solving Equations and Inequalities Containing Absolute Values We will express solutions in interval notation and graph on a number line. In the form of |x| < a and a > 0, then –a < x < a Example |2x| < 6 In the form of |x| > a and a > 0, then x < -a or x > a Example |3x - 3| > 15 In the form of |x| > a, a < 0, the solution set is all real numbers. Example |x| + 2 > -7 In the form of |x| < a, a < 0, the solution set is the empty set—ø Example |7x – 4| < -1 In the form of |x| > 0 or |x| < 0 Example |27x – 43| > 0 |27x – 43| < 0 In the form of |x+b| ≤ a|x+c|, thena . Then use test point method to solve. |x+c Example |x−2|<4x+4 | Practice Problems page 148 #54, 56, 60, 64
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