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# Study Guide Exam 1 MATH 1100.140

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This 16 page Study Guide was uploaded by Kait Brown on Friday September 23, 2016. The Study Guide belongs to MATH 1100.140 at University of North Texas taught by Mary Ann Barber in Fall 2016. Since its upload, it has received 20 views. For similar materials see College Algebra in Math at University of North Texas.

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Date Created: 09/23/16

Solve vs. Simplify o Solve Obtaining a value or a set of values, sometimes a graph or number line. An algebraic solution will include >, <, =, ≥, or ≤. o Simplify Obtain an equivalent expression in a simpler form Zero Product Property of Multiplication o If A•B=0, then A=0 and/or B=0 The Distributive Property of Multiplication Over Addition o A(B+C)=AB+AC DO NOT distribute multiplication over multiplication Importance of Notation/Hidden Meaning o => Implies o Є Element of o ⊂ Subset of o [ ] Includes endpoints o ( ) Does not include endpoints Types of Real Numbers and Their Importance o Natural N Counting numbers, starting at 1 {1, 2, 3…} o Whole W Counting numbers, starting with 0 {0, 1, 2…} o Integer Z Whole numbers, including negatives {…-1, 0, 1…} o Rational Q Any number that can be written as a ratio .872, ½, .3 o Irrational Q Any number that cannot be written as a ration, i.e., not a rational number √2, π P.1.1. 1. An integer that is not whole 2. An irrational number 3. Rational, but not an integer -3 π 1132/11 P.1.2. 5 1/7 -5 Z, Q, R Q, R R, Q, Z √4 √5 √-5 N, W, Z, Q, R Q, R Complex .123 . 123 .11235813… Q, R Q, R Q, R Set Notation o Set Collection of Objects o Roster Method A list of the elements {1, 2, 3, 4, 5, 6} o Set Builder Notation Describe the elements of the set {x x Є ℕ, x ≤6} OR {x x Є ℕ, x <7} o Number line How to show a set {x -1≤x<3} -2 -1 0 1 2 3 4 5 6 7 P.1.3. Write the set of natural numbers less than 10 with each set notation Roster Set Builder Number Line {1, 2, 3, 4, 5, 6, 7, 8, 9}{x x Є ℕ, x ≤9} <10} 1 2 3 4 5 6 7 8 9 10 11 12 13 Union and Intersection o Union ⋃ Represents the elements that are in one or both sets o Intersection ∩ Represents elements only in both sets P.1.4. If S={2, 3, 5, 7}, T={2, 4, 6, 8}, and V={1, 3, 5, 7}, find S ⋃ T and T ⋃ V S ⋃ T = {2, 3, 4, 5, 6, 7, 8} T ⋃ V = { } P.1.5. If A={x -1≤x<2} and B={x x>1}. Find A ⋃ B and A ∩ B. A={-1, 0, 1} -2 -1 0 1 2 A ⋃ B = {x -1≤x} B={ } 0 1 2 3 4 5 6 A ∩ B = {x 1<x<2} P.1.6. Use the number line below to write the set in interval notation. -5 -4 -3 -2 -1 0 1 2 3 4 5 [-4,1) ⋃ (2, 3] P.1.7. Rewrite │√17-5│without absolute value bars. 5-√17Because √17 is less than 5 P.1.9. Simplify 5(x-4)-10 1. Multiply 5 into (x-4) 5x-20 5 a. 5(x-4) 5x-20 2. Divide numerator by denominator, 5 a. 5x-20-10 (5x/5)-(20/5)-(10/5) x-4-2 3. Simplify a. -4-2 -6 b. x-6 Integer Exponents and Scientific Notation For a real number a, called the base, and a natural number n, called the exponent, aⁿ=a•a•a…a•a n number of times Exponential Rules to Know Product Rule o If a is a real number and m and n are integers, then a •a =a m n m+n o Keep base, add exponents Quotient Rule o If a is a nonzero, real number, and m and n are integers, then m n m-n a /a =a . o To divide exponents (with the same base), keep the base and subtract exponents. Power-of-a-Power Rule o If a is a real number and m and n are integers, (a ) =a m n mn o Keep base, multiply exponents Power-of-a-Product Rule n n n o If a and b are real numbers and n is an integer, (a*b) =a *b Power-of-a-Quotient Rule n n n o (a/b) =-n/b n n n o (a/b) =(b/a) =b /a 7 t P.2.4 Suppose x *x =1. What is t? When we multiply the same base with exponents, we add the exponents. So x •x =1. t=1/7 -2 P.2.5 Simplify 5 . To rid our equations of negative exponents we use the reciprocal of the 1 base, so the answer is 52 ( P.2.8 -2x3 -4 Simplify ( y 1 1. Apply -4 to the numerator: -2 x4 3(-4- 16 x-12And the denominator: y y-4 2. To get rid of the negative exponents we flip the numerator and the denominator: 1 - 16 x-12 y4 y-4 16x12 P.2.10. Write the following in scientific notation: 00.00365 3.65*10 -3 36500 3.65*10 4 Solve vs. Simplify o Solve Obtaining a value or a set of values, sometimes a graph or number line. An algebraic solution will include >, <, =, ≥, or ≤. o Simplify Obtain an equivalent expression in a simpler form Zero Product Property of Multiplication o If A•B=0, then A=0 and/or B=0 The Distributive Property of Multiplication Over Addition o A(B+C)=AB+AC DO NOT distribute multiplication over multiplication Importance of Notation/Hidden Meaning o => Implies o Є Element of o ⊂ Subset of o [ ] Includes endpoints o ( ) Does not include endpoints Types of Real Numbers and Their Importance o Natural N Counting numbers, starting at 1 {1, 2, 3…} o Whole W Counting numbers, starting with 0 {0, 1, 2…} o Integer Z Whole numbers, including negatives {…-1, 0, 1…} o Rational Q Any number that can be written as a ratio .872, ½, .3 o Irrational Q Any number that cannot be written as a ration, i.e., not a rational number √2, π P.1.1. 2. An integer that is not whole 2. An irrational number 3. Rational, but not an integer -3 π 1132/11 P.1.2. 5 1/7 -5 Z, Q, R Q, R R, Q, Z √4 √5 √-5 N, W, Z, Q, R Q, R Complex .123 . 123 .11235813… Q, R Q, R Q, R Set Notation o Set Collection of Objects o Roster Method A list of the elements {1, 2, 3, 4, 5, 6} o Set Builder Notation Describe the elements of the set {x x Є ℕ, x ≤6} OR {x x Є ℕ, x <7} o Number line How to show a set {x -1≤x<3} -2 -1 0 1 2 3 4 5 6 7 P.1.3. Write the set of natural numbers less than 10 with each set notation Roster Set Builder Number Line {1, 2, 3, 4, 5, 6, 7, 8, 9} {x x Є ℕ, x ≤9} <10} 1 2 3 4 5 6 7 8 9 10 11 12 13 Union and Intersection o Union ⋃ Represents the elements that are in one or both sets o Intersection ∩ Represents elements only in both sets P.1.4. If S={2, 3, 5, 7}, T={2, 4, 6, 8}, and V={1, 3, 5, 7}, find S ⋃ T and T ⋃ V S ⋃ T = {2, 3, 4, 5, 6, 7, 8} T ⋃ V = { } P.1.5. If A={x -1≤x<2} and B={x x>1}. Find A ⋃ B and A ∩ B. A={-1, 0, 1} -2 -1 0 1 2 A ⋃ B = {x -1≤x} B={ } 0 1 2 3 4 5 6 A ∩ B = {x 1<x<2} P.1.6. Use the number line below to write the set in interval notation. -5 -4 -3 -2 -1 0 1 2 3 4 5 [-4,1) ⋃ (2, 3] P.1.7. Rewrite │√17-5│without absolute value bars. 5-√17Because √17 is less than 5 P.1.9. Simplify 5(x-4)-10 1. Multiply 5 into (x-4) 5x-20 5 a. 5(x-4) 5x-20 2. Divide numerator by denominator, 5 a. 5x-20-10 (5x/5)-(20/5)-(10/5)x-4-2 3. Simplify a. -4-2 -6 b. x-6 Integer Exponents and Scientific Notation For a real number a, called the base, and a natural number n, called the exponent, aⁿ=a•a•a…a•a n number of times Exponential Rules to Know Product Rule m n m+n o If a is a real number and m and n are integers, then a •a =a o Keep base, add exponents Quotient Rule o If a is a nonzero, real number, and m and n are integers, then a /a =a m-n. o To divide exponents (with the same base), keep the base and subtract exponents. Power-of-a-Power Rule o If a is a real number and m and n are integers, (a ) =a m n mn o Keep base, multiply exponents Power-of-a-Product Rule o If a and b are real numbers and n is an integer, (a*b) =a *b n n n Power-of-a-Quotient Rule o (a/b) =a /bn n o (a/b) =(b/a) =b /a n n P.2.4 Suppose x *x =1. What is t? When we multiply the same base with exponents, we add the exponents. So 7 1/7 x •x =1. t=1/7 P.2.5 Simplify 5 .-2 To rid our equations of negative exponents we use the reciprocal of the 1 base, so the answer is 2 5 P.2.8 3 -4 Simplify ( -2x y -4 3(-4) 1 -12 3. Apply -4 to the numerator: -2 x - 16 x And the denominator: y y-4 4. To get rid of the negative exponents we flip the numerator and the denominator: 1 -12 4 - 16 x y y -4 16x 12 P.2.10. Write the following in scientific notation: -3 00.00365 3.65*10 36500 3.65*10 4 P.3.4. Find the product -6y(2y+5) *6y *6y 1. Multiply 6y through the equation 2 -12y -30y P.3.5. Find the product (x+3)(x +6x+7) (x(x +6x+7)) + (3(x +6x+7) 1. Reset equation. We want to multiply each part of the binomial by each part of the trinomial. (x +6x +7x)+(3x +18x+21) 2. Multiply x by the binomial and 3 by the binomial 3 2 x +9x +25x+21 3. Add like terms P.3.6. Simplify 2 2 (2x-1) -4x 2 (2x-1)(2x-1)-4x 1. Reset equation 2 2 (4x -2x-2x+1)-4x 2. Multiply (2x-1)(2x-1) (4x -2x-2x+1)-4x -4x+1 3. Combine like terms and simplify P.3.7. Multiply out the following: A) (x+5)(x+5) (x +5x+5x+25) x +10x+25 B) (x-5)(x-5) (x -5x-5x+25) x -10x+25 C) 2 2 (x+5)(x-5) x -5x+5x-25 x -25 P.3.8 Multiply out the following and simplify: A) (x+2) 3 (x+2)(x+2)(x+2) 1. Reset equation (x +2x+2x+4) 2. Multiply TWO binomials 2 (x+2)(x +4x+4) 3. Multiply last binomial into solution from step 2. 2 2 3 2 2 (x(x +4x+4))+(2(x +4x+4)) (x +4x +4x)+(2x +8x+8) x +6x +12x+8 4. Combine like terms. B)(x-2) 3 (x-2)(x-2)(x-2) 1. Reset equation 2 (x -2x-2x+4) 2. Multiply TWO binomials 2 (x-2)(x -4x+4) 3. Multiply last binomial into the solution from step 2. (x(x -4x+4))-(2(x -4x+4)) (x -4x +4x)-(2x -8x+8) x -6x +12x-8 4. Combine like terms. **Remember, when we’re subtracting polynomials, the signs can get confusing. Personally, it’s easier to switch all the signs on paper 3 before I start combining anything. So instead of working with (x - 4x +4x)-(2x -8x+8), I’m working with x -4x +4x-2x +8x-8. If I skip this step, or try to do it in my head, I always mess the entire thing up and have to start all over, convinced that I made an error in the very beginning. Don’t be like me. Be smart. 2 C) (x+4)(x -4x+16) 2 2 (x(x -4x+16)+(4(x -4x+16) 1. Multiply each term of the binomial in the trinomial (x -4x +16x)+(4x -16x+64) 2. Combine the resulting trinomials 3 2 2 3 x -4x +16x+4x -16x+64 x +64 3. Combine like terms to simplify 2 D)(x-4)(x +4x+16) (x(x +4x+16)-(4(x +4x+16) 1. Multiply each term of the binomial in the trinomial (x +4x +16x)-(4x +16x+64) 2. Combine the resulting trinomials 3 2 2 3 x +4x +16x-4x -16x+64 x -643. Combine like terms to simplify P.6.4 Find (x ) y (x ) =x3y P.6.5 3 y If (x ) =x, what is y? Facts that we know: (x ) =x =x 1 So what can you multiply by 3 to equal 1? x3(1/=x =x y=1/3 P.6.6 3 3 Describe in words, then find √x Cubed root of x raised to the power of 3. 3√x =(x ) =x =x1 P.6.8 Simplify the following: A) √-64=No real root 3 B) √-125=-5 1/4 1/4 (16/625) 16 2 625 1/4= 5 Simplify the incomplete problems below: √27= √300= 1. Factor the integer using a factor which is a perfect square √9*√3 √100*√3 2. Unsquare the perfect square. 3*√3 10*√3 P.6.10 Simplify the radical √(5/18) √5 √18 1. Rewrite the radical with √ on the numerator and the denominator √5 √9*√2 2. Factor denominator using perfect square √5 3*√2 3. Factor perfect square √5 * √2 = √10 = √10 4. Because we are dividing by √2, we want to multiply each side 3*√2 * √2 = 3*2 = 6 by √2, then simplify P.6.11 Simplify the radical √(7/5) √7 * 3√52 = 3√(7*25) 1. Reset the fraction and cross multiply by √5 (to even out) 3 3 2 3 3 √5 * √5 = √5 3√175 2. Simplify 5 Introduction to Chapter 1: Difference of Squares: a -b =(a-b)(a+b) Zero Product Property: ab=0; a, b=0 1.1 Linear Equations are first-degree equations in one variable that contain the equality symbol, “=”. Linear equation in the variable x can be expresses in the standard form, ax+b=c, where a ≠, b ∊ ℝ The domain is the largest set of acceptable input values. Of note, the denominator of a fraction cannot be equal to zero. 1.1.1 Solve ax+b=0, solve for x ax+(b-b)=(0-b) 1. Isolate x by subtracting b from both sides. x/a=-b/a 2. Divide both sides by a x=-b/a 1.1.2 Solve 5x-3=2x+8 -2x -2x 1. Subtract 2x from each side 3x-3=8 +3 +3 2. Add 3 to each side 3x=11 /3 /3 3. Divide each side by 3 x=11/3 1.1.3 Solve 400t-85=11-50t +50t +50t 1. Add 50t to each side 450t-85=11 +85 +85 2. Add 85 to each side 450t=96 /450 /450 3. Divide each side by 450 t=16/75 1.1.4 Solve 11-4(5+3x)+2(6x-7)=0 for x 11-4(5+3x)+2(6x-7)=0 11-20+12x+12x-14=0 1. Use distributive property 24x-23=0 2. Combine like terms x=-23/24 3. Simplify 1.1.5 2 Solve S=2πr +2πrh for h 2 2 S=2πrh +2πrh (S-2πrh )/(2πr)=h 1.1.6 x 1 5 − = 6 2 24 24( − =1 5 ) 6 2 24 1. Multiply all by LCD a. Divide each denominator by 24 and multiply that integer by the numerator x 1 5 17 24 −24 =24 4 x−12=5 4 x=17 x= ()6() ( 2 24 4 1.1.7 2x+1 +16=3x 1. Multiply all by LCD 3 1(2x+1)+3 (16=3(3x) a. Remember to first divide three by the denominator before multiplying it by the numerator 2x+1+48=6x 2x+49=6x 49=7x 2. Combine common terms 49/7=7x/7 x=7 1.1.8 x + 4 = 3 2 x −9 x+3 x −9 1. Find LCD by factoring out x -9 (It’s (x-3)(x+3) Solving Using the Quadratic Formula 2 -5x +7x+9 −b± √ −4ac −7± √9−4(−5)(9) x= 2a x= 2(−5) −7± 229 x= √ −10

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