The P/Q Mathematics Study Guide

The P/Q Mathematics Study Guide Copyright 007 by Lawrence Perez and Patrick Quigley All Rights Reserved

Table of Contents Ch. Numerical Operations - Integers... - Fractions... - Proportion and Percent... Ch. Variable Expressions - Integer Exponents... - Polynomial Expressions... - Rational Expressions... -4 Radical Expressions... Ch. Equations - Linear Equations... - Quadratic Equations... - Absolute Value Equations... -4 Rational Equations... -5 Radical Equations... Ch. 4 Inequalities 4- Linear Inequalities... 4- Quadratic Inequalities... 4- Absolute Value Inequalities... Ch. 5 Relations and Functions 5- Coordinate Geometry... 5- Linear Relations... 5- Systems of Relations... 5-4 Functions... Ch. 6 Geometry 6- Perimeter and Circumference... 6- Area... 6- Volume... 6-4 Angles in a Plane... 6-5 Special Triangles... 6-6 Trigonometry... Ch. 7 Probability and Statistics 7- Counting... 7- Probability... 7- Tables and Graphs... 7-4 Statistics... Copyright 007 by Lawrence Perez and Patrick Quigley All Rights Reserved

- INTEGERS Definition The set of integers, Z, consists of the whole numbers and their negative counterparts. Z = {, -, -, -, 0,,,, } Definition Example The absolute value of a number is the distance between the number and zero on a number line. x, if x 0 It is defined by the formula: x = x, if x < 0 Evaluate each absolute value below. a. 5 = 5 b. 5 = 5 c. 0 = 0 Property Example Multiplying Integers To multiply two integers, first multiply their absolute values. If the integers have the same sign then the result is positive, otherwise the result is negative. Multiply the integers below. a. = 6 b. ( ) = 6 c. ( ) = 6 d. ( ) ( ) = 6 Property Example Dividing Integers To divide two integers, first divide their absolute values. If the integers have the same sign then the result is positive, otherwise the result is negative. Divide the integers below. a. 0 = 5 b. 0 (-) = -5 c. (-0) = -5 d. (-0) (-) = 5 Property Example 4 Adding Integers To add two integers which have the same sign, add their absolute values and give the result the same sign as the integers. To add two integers with opposite signs, subtract the smaller absolute value from the larger and give the result the sign of the integer with the larger absolute value. Add the integers below. a. + 5 = 8 b. + (-5) = - c. (-) + 5 = d. (-) + (-5) = -8 Property Example 5 Subtracting Integers To subtract two integers change the subtraction symbol to addition while reversing the sign of the second integer. Then use the rules for addition. Subtract the integers below. a. - 5 = + (-5) = - b. - (-5) = + 5 = 8 c. (-) - 5 = (-) + (-5) = -8 d. (-) - (-5) = (-) + 5 =

- PROBLEM SET Evaluate each absolute value.. 7.. 0 4. 75 5. 6 6. 5 7. 7 8. Multiply the integers. 9. 7 0. ( 9). ( 6). ( 0) ( 5). ( 5) 4 4. ( 6) ( ) 5. 8 6. ( 8) 7. 6 ( 7) 8. ( ) 9 9. ( ) ( ) 0. 6 9 Divide the integers.. (-0) (-4).. 6 (-9) 4. (-9) 5. (-80) 0 6. (-5) (-) 7. 64 4 8. (-) 9. 4 (-7) 0. 08. (-7) 8. (-) (-) Add the integers.. (-) + 5 4. + (-6) 5. (-5) + (-8) 6. 5 + 7 7. (-) + (-5) 8. 9 + 9. (-) + 4 40. 4 + (-8) 4. + (-7) 4. (-6) + (-) 4. 8 + 6 44. (-) + 6 Subtract the integers. 45. (-) - 5 46. - (-6) 47. (-5) - (-8) 48. 5-7 49. (-) - (-5) 50. 9-5. (-) - 4 5. 4 - (-8) 5. - (-7) 54. (-6) - (-) 55. 8-6 56. (-) - 6 Evaluate. 57. ( - 9)( - 8) 58. ( - 7)(8-5) 59. 5(-) - 60. - (-)(-) 6. ( - 5)(-) 6. (4)(7-9) 6. 4-7 - 64. 8-9 - 65. 6 - (- + 5) 66. 8 - ( - 7) 67. 8 + ( ) 68. 7 - (80 6) 69. (-4)(5) (-) 70. (-5)() (6) 7. (0-4) (7 - ) 7. 5 + ( ) 7. ( 7) 4 74. 7 75. ( 0) 76. ( 9) + ( 7) 77. ( 4) ( 4)

- PROBLEM SOLUTIONS. 7.. 0 4. 75 5. 6 6. 5 7. 7 8. 9. 84 0. -7. -48. 50. -0 4. 66 5. 88 6. -4 7. -4 8. -08 9. 6 0. 54. 5. 4. -4 4. - 5. -4 6. 5 7. 6 8. - 9. -6 0. 9. -9.. 4. 7 5. - 6. 7. -7 8. 9. -7 40. -4 4. -6 4. -9 4. 4 44. 45. -8 46. 9 47. -7 48. - 49. 50. - 5. -5 5. 5. 8 54. - 55. 56. -9 57. 40 58. - 59. -6 60. 9 6. 9 6. -8 6. -5 64. - 65. 66. 4 67. 68. 69. 0 70. -5 7. -4 7. 7. 8 74. 4 75. 76. 6 77. 6

- FRACTIONS Properties Example If p q is a fraction and if r 0 then p q = p r q r and p q = p r q r. Reduce the fractions below. a. = 9 9 = b. 6 50 = 6 50 = 5 c. 4 49 = 4 7 49 7 = 7 Procedure Example Multiplying Fractions The product of the fractions p q and r s is p q r s = pr qs. Multiply the fractions below. Reduce your answers. a. 5 = 5 = 5 b. 4 = 4 = = = 6 Procedure Example Dividing Fractions The quotient of the fractions p q and r s is p q r s = p q s r = ps qr. Divide the fractions below. Reduce your answers. a. 5 = 5 = 5 = 0 b. 4 8 = 4 8 = 8 4 = 8 = 8 4 4 = Procedure Example 4 Adding Fractions The sum of the fractions p q and r s is p q + r q = p + r q. Add the fractions below. Reduce your answers. 0 + 4 = 0 + 5 4 5 = 0 + 5 0 = + 5 0 = 8 0 = 8 4 0 4 = 5 Procedure Example 5 Subtracting Fractions The difference of the fractions p q and r s is p q r q = p r q. Subtract the fractions below. Reduce your answers. 8 5 + 5 = 8 5 + 7 5 7 = 8 5 + 7 5 = 8 + 7 5 = 5 5 = 5 5 5 5 = 7

- PROBLEM SET Reduce each fraction... 7 5. 5 4. 5. 5 45 6. 8 7. 4 7 8. 00 6 9. 85 68 0. 56 0 Multiply. Reduce when possible.. 4. 7. 5 7 4 5 4. 50 5 7 5. 7 7 9 4 6. 4 7. 4 5 6 6 8. 5 44 Divide. Reduce when possible. 9. 4 0. 5 4 7. 6. 6. 0 5 4. 5 4 7 8 5. 60 7 6. 4 Add or subtract. Reduce when possible. 7. + 4 8. + 7 9. 5 + 4 5 0. +. 6 5 + 7. 0 + 5. 5 + 4. 6 + 5 5. 4 6. 7. 5 7 8. 4 5 5 9. 7 7 40. 0 6 4. 7 5 Perform the indicated operations. Reduce when possible. 4. 5 6 4. 5 + 4 + 44. 4 4 + 45. + 5 + 46. 5 4 5 47. 7 8 4 + 48. 4 5 6 8 9 49. 5 5 8 5 6 50. 4 5 0 5. +

- PROBLEM SOLUTIONS. 4. 5. 4 4. 5. 5 9 6. 7. 7 8. 50 9. 4 0. 5. 8. 7. 4. 70 5. 6 6. 7 7 7. 0 8. 9 75 9. 0. 5. -4.. 4. 5 8 5. 7 55 6. 4 5 7. 4 8. 9. 0. 5 6. 5.. 5 4. 7 5. 4 6. 6 7. 4 8. 5 9. 0 40. 4. 5 4. 4. 9 0 44. 45. 7 5 46. 47. 48. 5 49. 9 0 50. 0 5. 5 8

- PROPORTION AND PERCENT Definition A ratio is an ordered pair of numbers written x y where y 0. Theorem Definition Example Equality of Ratios a b = c if and only if ad = bc, where a, b, c and d are real numbers and b 0, d 0. d A proportion is a statement that two ratios are equal. Sam uses three tablespoons of instant cocoa to make two servings of hot cocoa. How many tablespoons of instant cocoa does Sam need if he is to make five servings of hot cocoa? Sam uses a ratio of tablespoons of mix to servings of hot water. Since Sam needs the same ratio of tablespoons of mix to 5 servings of hot water we get the following proportion. = x 5 where x is the number of tablespoons of mix required. Solving this equation for x: x = 5 x = 5 x = 7 tablespoons Definition Percent means parts per hundred. Thus n% represents the ratio n 00. Example What is 0% of 50? (0%)(50) = (0.0)(50) = 45 Example 5 is what percentage of 80? 5 80 = 0.08 = 8. % = 8 % Example 4 Suppose apples cost $.5 per pound. If oranges cost 0% more than apples, how much will three pounds of oranges cost? We first calculate 0% of $.5: (0%)($.5) = (0.)($.5) = $0.. Therefore each pound of oranges cost cents more than each pound of apples. This implies that the cost of pound of oranges is $.5 +. = $.8. We conclude pounds of oranges costs, ($.8) = $4.4.

- PROBLEM SET Find the ratio of each pair of quantities.. 8 feet to 4 inches. gallons to quarts. 40 seconds to minute 4. $.65 to 5 5. hour and 5 minutes to 5 hours In each of the following proportions, find the value of x. 6. 5 = 5 x 7. x 49 = 9 8. x = 5 9..5 4 = 8.4 x 0. 7 = x. x 5 = 4. 9 = x 4. x = 5 4. x = b 5. 9 x = x 6 6. r s = c x 7. b x = b Solve each proportion problem. 8. Candy bars sell at a rate of for 4. What will 0 candy bars cost? 9. A wheelbarrow can carry 5 lbs in 7 loads. How many trips are needed to haul 40 lbs? 0. If three painters could paint four apartments in a day, then how many apartments could five painters paint in a day?. 55 miles per hour is approximately 88 kilometers per hour. If a car travels at 90 miles per hour, then approximately how many kilometers per hour is it traveling?. The scale on a map is inches to 6 miles. How far apart are two cities if the 4 map shows them as 4 inches apart? Solve each percentage problem.. What is % of 00? 4. 8 is what percentage of 400? 5. A jacket regularly costs $04.0. If it is on sale at 0% off then how much does it cost? 6. If a realtor earns a 5% commission on every house that she sells, then how much will she earn on the sale of a $,000 house? 7. At Albert Einstein Junior High School there are students who participate in sports. If this represents 5% of the student body, then how many students attend Albert Einstein JHS?

- PROBLEM SOLUTIONS. 4. 0. 4. 5. 4 6. 5 7. 8. 9..4 0. 9... 5 4. 6 b 5. ± 6. cs r 7. b 8. $.40 9. 9 trips 0. 6. 44 kph. miles. 56 4. 7% 5. $8.6 6. $650 7. 80 students

- INTEGER EXPONENTS Definition a raised to the nth power, written as a n, means a multiplied times itself n times, a n = a a a a, where a is called the base and n is called the exponent. n factors of a The expression 4 is read to the fourth power and means = 8. Properties Properties of Exponents Examples a m a n = a m +n x x = x 5 a m a = x 7 n am n = x x 4 (a m ) n = a mn (x ) = x 6 a b m = a m b m a m = Example Simplify the expression (x y ). a m (x y ) = (x y ) = (x ) (y ) = 4x 4 y 6 x y x = 4x y = x m a = b m x b a m = y y 4x a 0 =, where a 0 ( x ) 0 =, where x 0 = y6 4x 4 Example Simplify the expression x y x y = = x y x y y x x y x y x y. = y x 5

Example Simplify the expression x xy. x xy = x x y Example 4 Definition Example 5 = y x x = y x Simplify the expression x y z. (xy z ) x y z = x y z (xy z ) x y z = xy y x z z = xy5 x z A number is in scientific notation when it is written in the form a 0 n a <0 and n is an integer. Convert,000 to scientific notation. where,000 =. 0,000 =. 0 4 Example 6 Convert -0.005 to scientific notation. -0.005 = -.5 000 =.5 0 =.5 0 Example 7 Convert 6.5 0 4 to decimal notation. 6.5 0 4 = 6.5 0,000 = 6,500 Example 8 Convert 8.4 0 5 to decimal notation. 8.4 0 5 = 8.4 0 5 = 8.4 00,000 = 0.000084 Example 9 Multiply (.5 0 5 )(4.0 0 8 ) and express your answer in scientific notation. (.5 0 5 )(4.0 0 8 ) = (.5 4.0)(0 5 0 8 ) = 0 0 =.0 0 4

- PROBLEM SET Simplify each expression.. (a b)(ab 4 ). ( x y). [(x) ] 4. (rs 4 )(r s) 4 5. x n x n 6. (r s ) 7. a 5 a 8. x y 4 x 4 y 9 9. (a b ) 0. (a b) x y x y. ( xy ). ( xy ). (x y z ) 4. [(a b ) ] x z 5. 4a b c (ab c ) 6. (a c ) (a c ) 7. a a 4 8. (c ) (c ) 9. (x y) (x y) 0. 4x + y ( x). (a + b). a + (a + b) b b. ( x ) y ( y ) x 4. (x) 0 (a + b), x 0 5. [( ) ] 6. x(x + x) 7. (x y ) ( ) 8. (a + b 0 ) 6 (a +) 5, b 0 Express each quantity in scientific notation. 9. 500 0. 0.004. 4,6,000. 0.000056. 4,000,000,000 4. 40.04 5. -. 6. -8 7. -0.004 Express each quantity in decimal notation. 8. 0 9. 0 4 40..5 0 5 4. 9.75 0 4. 4. 0 8 4. 0.00 0 44. 8.75 0 0 45..6 0 46..6 0 Perform the indicated operation and express your answer in scientific notation. 47. ( 0 5 )( 0 7 ) 48. 0 4 49. (. 0 4 )(5 0 6 ) 4 0 50. 0.0000049 0.00007 5. 0.0 0 7.0 0 5. (5,000,000)(0.0000)

- PROBLEM SOLUTIONS. a b 5. 7x 6 y. 64x 6 4. r 9 s 8 5. x n 6. 9. 4a 4 b 0. s r 7. 8. 6 a y x 6. 8x y 6. x 7 y 5 4x y 4. z 6 8y 4 4. 4b 4 5. a 4b 7 6. c 5 c 4 8a 7. a 8. c 4 9. (x y) 5 0. y. a + b. a. 4. a + b 5. - 6. + x 7. xy y x 8. a + 9..5 0 0. 4 0. 4.6 0 6. 5.6 0 5. 4 0 9 4. 4.004 0 5.. 0 6..8 0 7..4 0 8. 000 9. 0.000 40. 5,000 4. 0.00975 4. 4,000,000 4. -0.00000 44. -8.75 45. -60 46. -0.006 47. 6 0 48. 0 6 49..6 0 9 50. 7 0 5. 4 0 5. 0

- POLYNOMIAL EXPRESSIONS Definition Example Terms with exactly the same variable parts (the same variables raised to the same powers) are called like terms. Terms with different variable parts are called unlike terms. Simplify the polynomial expression 7x + xy 4x + xy y by combining like terms. 7x + xy 4x + xy y = 7x 4x + xy + xy y = (7 4)x + ( +)xy y = x + 4xy y Example Given x = - and y =, what is the value of xy xy. Procedure xy xy = ( )() ( )() = ( )(9) ( 6) = -8 + = -6 Multiplying Polynomials Two polynomials are multiplied together by multiplying each term of the first polynomial by each term of the second polynomial and then summing the products. Example Find the product of (x y)(x + y ). (x y)(x + y ) = (x )(x) + (x )(y ) (x )() (y)(x) (y)(y ) + (y)() = x + x y x xy y + y Formulas Description Formula Difference of two squares a b = (a b)(a + b) Perfect Square Trinomials a + ab + b = (a + b) a ab + b = (a b) Sum of two cubes a + b = (a + b)(a ab + b ) Difference of two cubes a b = (a b)(a + ab + b ) Example 4 Factor 9x 64y. Since the expression 9x 64y is the difference of two squares we have 9x 64y = (x) (8y) = (x 8y)(x + 8y) Example 5 Factor 6m 6mn + 4n 4. 6m 6mn + 4n 4 = 4[4m 4mn + n 4 ] = 4[(m) (m)(n ) + (n ) ] = 4(m n )

- PROBLEM SET Evaluate each polynomial for the given values.. x + where x =. x where x = -. x +x where x = - 4. 4t 6t + where t = 5. x y x + where x = - and y = 6. ab b + where a = and b = - Simplify by combining like terms. 7. (7x )+ (5x + ) 8. (y + 4) (4y ) 9. (x x + 7) + (x + 4x ) 0. (5x + x ) (x 4x ). (a b + 5ab ) +(ab + ab a b). (5b ab + 4a ) (a b ab). (x x + 5) + ( x + x + ) 4. (xy + y ) (x y + x + y) Multiply and then simplify. 5. 4x (x 9) 6. 4y (y 5y + 7) 7. 4a (5b b + 4) 8. (x + )(x ) 9. (x )(x + x ) 0. (t 6)(t t + ). (x 5)(x +). (r r + 7)(r + r ) Factor completely.. x 9 4. 64r 5s 5. 4x +6 6. 98x 50x 7. 9x 60x +00 8. 5x + 40x +6 9. 7x 4 + 7x + 48x 0. 45a + 60a + 0a. 64x 80xy + 5y. 6a 4 7a b + 8b 4. x + 4 4. x + 8 5. x 7 6. 5x 5 + 40x 7. 4t 4 t 8. y + 7y 9. 64r + 7s 40. a +8b 6

- PROBLEM SOLUTIONS. 5. -4. - 4. 0 5. 6 6. -8 7. x + 0 8. -y + 6 9. 4x + x + 5 0. x + 7x +. a b + ab + 6ab. 5b + a + b. 6 4. xy x y x 5. 8x 6x 6. y 4 0y + 8y 7. 0a b a b +6a 8. x + x 9. x 5x + 0. t t + 7t 54. x x 9x 5. r 4 + r r +7r. (x )(x + ) 4. (8r 5s)(8r + 5s) 5. 4(x + 4) 6. x(7x + 5)(7x 5) 7. (x 0) 8. (5x + 4) 9. x (x + 4) 0. 5a(a + ). (8x 5y). (a + b) (a b). x + 4 4. (x + )(x x + 4) 5. (x )(x + x + 9) 6. 5x (x + )(x x + 4) 7. 4t(t )(t + t + 4) 8. y(y + 9) 9. (4r + s)(6r rs + 9s ) 40. (a + b )(a ab + 4b 4 )

- RATIONAL EXPRESSIONS Definition A rational expression has the form P, where P and Q are polynomials and Q 0. Q Property If P Q is a rational expression and if R is a polynomial such that R 0, then PR QR = P Q. Example Procedure Example Simplify the rational expression: x 0 x 0x + 50 = (x 5) (x 0x + 5) (x 5) = (x 5) = x 5 x 0 x 0x + 50. Multiplying Rational Expressions The product of the rational expressions P Q and R S is P Q R S = PR QS. Multiply the following rational expressions: + x x +. Simplify your answer. + x x + = x x + x x + = = x ( x +) x(x +) = x x + x + Procedure Example Dividing Rational Expressions The quotient of the rational expressions P Q and R S is P Q R S = P Q S R = PS QR. Divide the following rational expressions x x x x x = x(x ) = x (x ) = x (x ) x. Simplify your answer. x

Procedure Example 4 Procedure Example 5 Adding Rational Expressions If P Q and R Q are rational expressions, then P Q + R Q = P + R Q. Add the following rational expressions: x x + x = x (x )(x +) + x x = (x )(x +) + (x ) x (x +) = (x )(x +) x x = (x )(x +) x = (x )(x +) = x + x x +. Simplify your answer. x (x +) (x +) Subtracting Rational Expressions If P Q and R Q are rational expressions, then P Q R Q = P R Q. Subtract the following rational expressions: x x 6 x + 4x. x x 6 x + 4x = (x )(x + ) (x )(x + 7) (x + 7) = (x )(x + )(x + 7) (x + ) (x )(x + 7)(x + ) (x + 7) (x + 4) = (x )(x + )(x + 7) x + 7 x 4 = (x )(x + )(x + 7) (x ) = (x )(x + )(x + 7) = (x + )(x + 7) = x + 9x + 4

- PROBLEM SET Simplify each expression.. 8xy x. (a + )(a ) (a + )(a + ). x 6 9x 4. x + x x 5. h 6 h + 4 6. x + x x + 7. c + c + c 8. z + 4z z z 9. (t )(t + ) 4t 4t + 0. x + 6x + 9 x + 7. 9x 4 7x 8. 9m 6 7m 64 Perform the indicated operation and simplify the result.. a y x a 4. r s t s 5. x y y 4x 6. 4b 9a 7a 7. (t ) t + t t + 8. x + x x x x 9. 6y y 6y 6y + 0. t + t +t t + t +. 4k 8 k + k 0 k +. 5x + x x 5x + 4x 8. a + ab b b a ab 4. 0 x + x + x 4x + 6 5. 9x 4 7x 8 x + x 6. x x 4x 9 x x + 7. x +0x + 5 x x 4 x +0 8. a + b 9. 5 x + + x x + 0. 9 x + 5 x +. 9 r + 8 r. x z + z y. 5 xy 4. 8 m + + m 5. b b + 4 b 6. n n + + n + 7. z z + 4 z 8. 0 x y y x 9. x + x 40. t t + + t 4. x x 5 x + 4. y + y 4 + y + y + 4 y 8

- PROBLEM SOLUTIONS. y. a a +. x x 4. x + 5. h - 4 6. x 7. c + c 8. z + 4 z 9. t + t 0. x + x + x + 9. x + 9x 6x + 4. m + 4 9m m +6. x y 4. r t 5. y 6 6. b a 7. t + 8. x + 9. y 4 y + 0. t. k 4 k 5. 4x. a + b a b 4. 5x x + 5. x 9x 6x + 4 6. x x 7. x +5 8. a + b ab 9. x + 5 x + 0. 4 x +. 7 r. z + xy yz. 5 xy xy 4. 0m + 6 m + m 5. b + b 8 b 5b + 6 6. 7. z 4 z 8. x y 9. 5x 6 x x 40. t + t + 4 t + t 4. x x + 5 x 4. y

-4 RADICAL EXPRESSIONS Definition The nth root of a number a is a number that when raised to the nth power, produces the n number a. It is written as a, where n is the index, is the radical and a is the radicand. For example, 9 = since = 9 and x 6 = x since (x ) = x 6. 4 Similarly, 6 = since 4 = 6 and x = x 4 since (x 4 ) = x. Definition Rational exponents are exponents of the form m, where m is the power of the number n and n represents the nth root of the number. Thus a mn = n a m n = a ( ) m where a 0. For example, 7 = 7 = and 8 = ( 8) = () = 4 Example Write the expression x in radical form. The expression x can be written as ( x ) or ( x ). Therefore, x can be written in radical form as ( x) or x. Example Simplify the expression 8 8. 8 8 = 9 4 = = ( ) = Example Simplify the expression x 7 y 5 z 6 x 0 y 8 z. x 7 y 5 z 6 x 0 y 8 z = x 6 x y y z 6 x 9 x y 6 y z = (x yz ) xy (x y z) xy = (x yz x y z) xy = x yz(z xy) xy Example 4 Multiply ( ) ( 8 4). ( ) ( 8 4) = 8 4 8 + 4 = 6 8 4 + = 4 4 4 6 + 4 = 4 6 +

-4 PROBLEM SET Write each exponential expression as a radical expression.. 5. (x). (xy ) 4 4. 5x 5. (a b 4 ) 5 Write each radical expression as a exponential expression. 6. 7. 5 a 8. y 9. 4x x 0. r 4 s Simplify each radical expression.. 64. 64. 4. 7 5. 4 9 6. x 7. a b 4 8. 8x 6 y 9 9. 4 6r 8 5 0. x 5 y 0. 7a 9 5. 4x 5 y 5. 5x 4. y 4 7a b 5 5. (x + y) 6. (x + y) 6 7. 5 + 5 8. 5 5 9. 0. a 7 b 7. 6. x x x. x + x 4. x x 5. 6. x + y x + y 7. 6x x 8. 9. x x 40. x 8x 4. 64 4. x 4. x 44. 4 ab 45. 5 x y 5y x 46. y 4 4x x y y 47. 5 x y 5y x 48. y 4 4x x y y 49. 4 + 50. 4 5. 7 + 7 5. 5. + 54. 7 7 55. ( 0+ 5) 56. ( 8 6) 57. 58. + 59. + + 60. + 5 5

-4 PROBLEM SOLUTIONS. 5. x. y 4 x y 4. 5 x 5. 5 a b 4 6. 7. a 5 8. (y) 9. 4x 0. ( 74 )(r)(s 4 ). 8. 4. - 4. - 5. 6. x 6 7. a 6 b 8. x y 9. r 0. xy. a. x y 5. 5x y 4. a 4 5. x + y b 5 6. (x + y) 7. 5 5 8. 5 9. 0. (a b) 7. 5. x x. x 4. x 5. 6 6. x + y 7. x 8 8. 9. x 40. x x 6 4. 4. x 4. x 44. 4 ab 45. 5 y xy 46. x 8y 47. x y y 48. y x 8x 49. 7 50. 5. 6 5. 5. 4 54. 5 6 55. 5 + 0 56. 6 57. + 58. 59. 4 60. + 5 5

- LINEAR EQUATIONS Definition Property Property Linear equations in one variable are equations which can be written in the form ax + b = 0 where a and b are real numbers. The Addition Property of Equations If a = b and c is a real number then a + c = b + c. The Multiplication Property of Equations If a = b and c is a real number then ac = bc. Example Solve the equation for x: 5x + 5 = 0. 5x + 5 = 0 5x + 5-5 = 0-5 5x = -5 5x 5 = 5 5 x = - Example Solve the equation for x: 4x + 5 = 6. 4x + 5 = 6 4x + 5-5 = 6-5 4x = 4x 4 = 4 x = 4 Example Solve the equation for x: x - 5 = x +. x - 5 = x + x - 5 - x = x + - x x - 5 = x - 5 + 5 = + 5 x = 6 x = 6 x = Example 4 Solve the equation for x: ax + b = c. ax + b = c ax + b - b = c - b ax = c - b ax a = c b a x = c b a

Example 5 Solve the equation for x: x = 5 6 x +. x + = 5 6 x + + x = 5 6 x + 4 6 x 5 6 x = 5 6 x + 4 5 6 x 6 x = 4 x = 4 ( ) x = 4 ( ) x = -4 Example 6 Solve the equation for x: -6x + 7 = x -. -6x + 7 = x - -6x + 7-7 = x - - 7-6x = x - 9-6x + 7 - x = x - - x -8x = 9 8x 8 = 9 8 x = Example 7 Solve the equation for x: (x ) = x +. (x ) = x + x 6x + 9 = x + x 6x + 9 x = x + x -6x + 9 = -6x + 9-9 = - 9-6x = -6 6x = 6 6 6 x = Example 8 Three more than twice a number x is equal to y. Solve the equation for x. x + = y x + - = y - x = y - x = y x = y

- PROBLEM SET Solve each equation.. 4x = 6. x + = 64. x - 6 = 4 4. 7. x 4 = 8 5. x = 6 6. x 9 5 = 40 5x = 0 8. x + 57 = 8 9. 4 = x - 4 0. 5(b - ) = 7. 4 + 6(r + ) = 9. x - = 8x - 9. 6s + 8 = 8-5s 4. + 9x 6 = 5. 7 s = 6. x + = 7. y 4 + 7 = 0 8. t 0 = 6 9. 0.c +.5 = 0.8c 0. 0.(v - 8) = 0. w 5 6 = w. 4.7x -. = 9.8. t + 8 = 5 t 4. 6(k + 0) = 5(k + 4) 4 5. t - 7 = 8t + 5 6..8 = 7x - 4.9 7. r + = r + 5 6 8. 8-9a = 9a - 8 9. 5 (t 4) = t 0. 4( - b) = (b + 4) 5. 5x + = x. a + = a 5 4. 5 x = x 6 4. x = 4 5. x = 4 6. x = 5 Solve for x. 7. ax + cd = h 8. a(b + x) = d 9. r + x = 6 40. 5x - 6 = q + 4 4. 8 - x = r - 7 4. 5(x + s)= x Write an equation that represents each word statement. Solve the equation for x. 4. Two more than three times a number x is equal to 7. 44. Two more than three times a number x is equal to. 45. Three less than four times a number x is equal to y.

- PROBLEM SOLUTIONS. 9. 5. 49 4. 5. 9 6. 5 7. 4 8. 5 9. 6 0. 5. 7 6.. 0 4. 70 9 5. 6. 6 7. 4 8. 9. 0. 08. 5 7. 50 9. 96 7 4. 0 5. 5 6. 0.86 7. 8. 8 9 9. 5 0. -4. 6. 5 7. 5 4. 6 5. 9 8 6. 0 7. h cd a 8. d ab a 9. 6 r 40. q +0 5 4. 5 r 4. 0s 4. x + = 7; x = 5 44. x + = -; x = 5 45. 4x - = y; x = y + 4

- QUADRATIC EQUATIONS Definition Quadratic equations in one variable are equations which can be written in the form ax + bx + c = 0 where a, b and c are real numbers and a 0. Quadratic equations can have two real solutions, one real solution or no real solutions. They can often be solved by factoring and applying the zero-product property. Property The Zero-product Property For any two real numbers a and b if ab = 0 then a = 0 or b = 0. Example Solve the equation for x, x = x +. x = x + x x = 0 (x )(x +) = 0 x - = 0 or x + = 0 x = x = - Example Solve the equation for x, x 6x + 4 = x + 6x 5. x 6x + 4 = x + 6x 5 4x x + 9 = 0 (x ) = 0 (x ) = ± 0 x - = 0 x = x = Example Solve the equation for x, x +8 = 0. x +8 = 0 x = -8 x = -9 x = ± 9 x = ±i Thus there are no real solutions.

For some quadratic equations it is necessary to use the quadratic formula. Formula The Quadratic Formula For any quadratic equation ax + bx + c = 0, x = b ± b 4ac a. Example 4 Solve the quadratic equation for x: x +x 5 = 0. x = b ± b 4ac a = () ± () 4()( 5) () = ± 49 4 = ± 7 4 So x = + 7 4 = 4 4 = or x = 7 4 = 0 4 = 5. Example 5 Solve the quadratic equation for x: x + x 5 = 0. x = b ± b 4ac a = () ± () 4()( 5) () = ± 9 4 So x = + 9.9 or x = 9 4.9. Example 6 Solve the quadratic equation for x: x x + = 0. x = b ± b 4ac a = ( ) ± ( ) 4()() () = ± 4 4 = ± i 4 = ±i So x = + i or x = i. Example 7 The sum of three times x and twice the square of x is equal to seven. Solve the quadratic equation for x. x +x = 7 so x +x 7 = 0. x = b ± b 4ac a = () ± () 4()( 7) () = ± 5 4 So x = + 5 4 or x = 5 4.

- PROBLEM SET Solve each quadratic equation by factoring.. y 7y + 70 = 0. x + 9x + = 7. x(x +) = 5x 4. a + 5 =0a 5. d + 5d = 6. a + a + = (a + ) 7. 0 9y = y 8. x = 5x 9. c(c + 4) = +(9 + c) 0. a = 4(a ). b(b + ) =. a(a + 6) = 5 a(a + ). (x + ) = 6(x + 5) 4. 0(0 + y ) 9(y + ) (+ y ) = Solve each quadratic equation using the quadratic formula. 5. x + 7x 5 = 0 6. x x 4 = 0 7. t + 4t + = 0 8. w + w + 4 = 0 9. w + w = 0 0. k + 5k + = 0. x 4x + 4 = 0. 9x + 6x + = 0. 4x x = 0 4. x + 4 = 0 5. x = 5x + 0 6. (x + )(x +) = 7 7. x = + x 8. x + 6 x = 0 9. 0.5t = t + 0.5 Solve each quadratic equation using any method. 0. 9x +x + 4 = 0. 9x + = 0. x + 5x + = 0. x 6x + 5 = 0 4. x = 5(x ) 5. x x = 4 x + x 6. x 4 + = x 7. x 4x + 45 = 0 8. x + 5x + = 0 9. 9(x ) = 0 40..9x = 4.x 9 4. x 4 + x + x = 0 4. x + 7x 6 = 0 4. x + 7 6 x = 44. (x )(x + ) = 4 45. 4x x = + x 48. 4x 49 = 0 49. x + 5x = 0 50. 46. 5x +5x = 0 47. (x )(x + 4) + 6 = 4x x + x x x + = 8

- PROBLEM SOLUTIONS. { 7, 0 }. { -5, -4 }. { -4, 8 } 4. { 5 } 5. 4, 6. { -4, - } 7., 5 8., 9. { -6, 5 } 0. {, 6 }. { -, - }. 5,. { -9, 9 } 4. { -9, 9 } 5. 5, 6., 7 7. { ± } 8. { ± i } 9. ± 7 4 0. 5 ± 6. { }.. { 0, } 4. { ± i } 5. 5 ± i 55 6. ± 57 4 7. ± 5 8. ± 7 6 { } 0. 9. ±. ± i. 6 ±.,5 4. 5 ± 5 5., 6. {, 4 } 7. { 5, 9 } 8. 4, 9., 4 40. 9 ± 5,84 86 4. 4, 4., 4., 44. { -4, 5 } 45. {, 4 } 46. { -, 0 } 47. 4, 48. ± 7 49., 50. ± 6

- ABSOLUTE VALUE EQUATIONS Theorem If u = c where u is a variable expression and c is a non-negative real number, then u = c or u = -c. Example Solve the equation for x: x = 6. x = 6 or x = -6 Example Solve the equation for x: x + = 5. x + = 5 or x + = -5 x = x = -8 Example Solve the equation for y: y 4 = 7. y - 4 = 7 or y - 4 = -7 y = y = - y = y = Example 4 Solve the equation for x: x + = 7. x + = 7 x + = 0 x + = 0 or x + = -0 x = 8 x = - x = 6 x = Example 5 Solve the equation for t: t + = 7. t + = 7 or t + = -7 t = 5 t = -9 t = ± 5 t = ± 9 t = ± 5 t = ±i So t =± 5 are the only real solutions.

- PROBLEM SET Solve each equation. If there is no solution then write no solution.. x = 5. x = 0. v = 5 4. x + = 5 5. p = 4 6. x = 7. t = 8. t = 9. y = 5 0. w = 4. x 4 =. 5x + = 0. a a = 4. x + =5 5. x + = 8 6. 7z + = 7. 4 a 4 = 8. x + 4 =6 9. + q = 5 0. x + + 6 = 0. x = 4. x + 4 = 4. 54x + 7 = 5 4. t = 4 5. n + 4 = 6. 4x 9 = 0 7. y 5 = 8. 6y = 4 9. x 5 = 0. x = 5x. t = t 4. 4 5 r + 9 =. h 4 7 = 4. 4s 5 = 7. y = 5 6. x + = 4 7. p = p 4 8. 4 b = 9. x + = 40. t 5 = 4 4. 8 s = 4. a 7 +8 = 0 Solve for x. 4. ax + b = y 44. ax+ b = y 45. x + y = 4

- PROBLEM SOLUTIONS. { -5, 5 }. { 0 }. 5 6, 5 6 4. { -, } 5. 4, 4 6. no solution 7. { -4, 4 } 8. { -4, 4 } 9. { -5, 5 } 0. { -, }. {, 6 }. 5, 9 5., 4., 9 5. 7, 9 6. 7, 7 7. 4, 5 8. { -0, 8 } 9. 4 5, 5. 0, 5. 7 6, 0. 5,., 6. ±., 5 4. { -8, 8 } 7. {, 8 } 8. 7 6, 6 9., 7 0. 5 ± 5 ±,., 5 4. 65 6, 5. {, 7 } 4. 8, 8 7. { -, 8 } 6. { -4, 0 } 7. 4, 4 8. {, 4 } 9. { -7, 4 } 40. no solution 4. 8, 0 4., 4. y + b a, y b a 44. ± y b a 45. 4 + y, 4 y

-4 RATIONAL EQUATIONS Definition Rational equations are equations involving rational expressions. The technique for solving rational equations is called clearing the fractions. Procedure Clearing the Fractions. Factor all polynomial expressions.. Multiply through by the common denominator of all the terms.. Reduce wherever possible 4. Solve the resulting equation 5. Check your answers using the original equation Example Solve x + x + x = x Step x + x + x = x x + x(x +) = x Step x(x + ) x + x(x +) x(x +) = x(x +) x Step x - = (x + ) Step 4 x - = x + x - x = + -x = 4 x = -4 Step 5 x + x + x? x ( 4) + ( 4) + ( 4)? ( 4)? 4 8? 4 9? 4 4 = 4 So -4 is the solution of x + x + x = x.

Example Solve x 5x + 5 = x + + x + x +. x 5x + 5 = x + + x + x + x = 5(x +) x + + (x +)(x + ) x 5(x +)(x + ) = 5(x +)(x + ) 5(x + ) x + + 5(x +)(x + ) (x + )(x + ) So - is not a solution of x(x + ) = 5(x + ) + 5 x + x = 5x + 5 + 5 x + x = 5x + 0 x x 0 = 0 (x + )(x - 5) = 0 x + = 0 or x - 5 = 0 x = - x = 5 So 5 is the only solution of x 5x + 5? x + + x + x + ( ) 5( ) + 5? ( ) + + ( ) + ( )+ 5? 0 + 0 5 undefined x 5x + 5 = x + + x + x +. x 5x + 5? x + + x + x + (5) 5(5) + 5? (5) + + (5) + (5) + 5 0? 7 + 4 6? 6 4 + 4 6? 7 4 6 = 6 x 5x + 5 = x + + x + x +.

-4 PROBLEM SET Solve each equation for x.. a + b = c + a x. a x = b. a x + = x 4. a + b = c x Solve each rational equation. If there is no solution then write no solution. 5. x + 5 = x + 6. a + 5 a =0 7. 4 b 4 b = 8. t 6 = 0 9. x = 8 x 9 0. 5 x x + = x 4. y = + y. 5 x 7x + = x + 5 x 4. 4 y 4 y = 4. z 5 = 0 + z 0 5. b = 0 b 6. x 6 + x + 8 = 0 7. x + x 4 = x 6 8. x x 4 + 6 x = 6 (x 4)(x ) 9. 8 a + = 9 a 0. x + + x = x. x x =. 5 x = x 4. x + = 4 x 4. x x + x + = x 5. p = + p 6. 4t 9 + t = t + 7. x + x(x ) + x(x )(x ) = 0

-4 PROBLEM SOLUTIONS. a + c a + b. ab b a. - a 4. ab b a 5. { -6 } 6. 5 7. {, 6 } 8. ± 4 9. { 5 } 0.. { }. no solution. {, 6 } 4. 4 5. { -, 5 } 6. - 7. { 0 } 8. { -8, 5 } 9. { 8, } 0. 6. { -, 4 }. SKIP. { - } 4. no solution 5. 6. 7 7. { - }

4- LINEAR INEQUALITIES Definition Properties Properties Linear inequalities in one variable are inequalities which can be written in one of the following forms: ax + b > 0 ax + b < 0 ax + b 0 ax + b 0 where a and b are real numbers. The Addition Properties of Inequalities If a > b and c is a real number then a + c > b + c. If a < b and c is a real number then a + c < b + c. The Multiplication Properties of Inequalities If a > b and c is a positive real number then ac > bc. If a < b and c is a positive real number then ac < bc. If a > b and c is a negative real number then ac < bc. If a < b and c is a negative real number then ac > bc. Example Solve the inequality, 5x + 5 > 0. 5x + 5 > 0 5x + 5-5 > 0-5 5x > -5 5x 5 > 5 5 x > - -7-6 -5-4 - - - 0 4 5 6 7 Example Solve the inequality, 4x + 5 6. 4x + 5 6 4x + 5-6 6-5 4x 4x 4 4 x 4-0.5.5.75

Example Solve the inequality, -x - 4 < 0. -x - 4 < 0 -x - 4 + 4 < 0 + 4 -x < 4 x > 4 x > - -7-6 -5-4 - - - 0 4 5 6 7 Example 4 Solve the inequality, -6x + 7 x -. -6x + 7 x - -6x + 7-7 x - - 7-6x x - 9-6x - x x - 9 - x -8x -9 8x 8 9 8 x - - - 0.5 Example 5 Solve the inequality, (x ) x +. (x ) x + x 6x + 9 x + x 6x + 9 x x + x -6x + 9-6x + 9-9 - 9-6x -6 6x 6 6 6 x -7-6 -5-4 - - - 0 4 5 6 7

4- PROBLEM SET Solve each linear inequality and graph the solution on a number line.. x > 6. x + 4 < 64. x - 7 4 4. x 8 5. y - < -4 6. -x > 4 4 x 7. r + 4r - 5 8. 0. - t < 7. (m - 5) -9. 6 9. -(x - ) > 8 x 9 5 40. 5x < 0 4. x + 57 > 8 5. 4 x - 4 6. 5(b - ) 7 7. x + > 8. x - < 8x - 9 9. -k - 0. x + 5. 7 s <. 6s + 8 > 8-5s. + 9x 6 4. 6s + 4s - 5. 4 + 6(r + ) > 9 6. 7 - x < 7. 8. 0.c +.5 0.8c 9. 0.(v - 8) < 0 0. t 0 6 w 5 6 > w. 4.7x -. 9.8. t + 8 5 t. 6(k + 0) > 5(k + 4) 4 4. t - 7 < 8t + 5 5..8 7x - 4.9 6. r + r + 5 6 7. s + < 7s - 5 8. 40. 8-9a 9a - 8 4. y 4 + 7 > 0 9. - (7 - x) ( - x) 5 (t 4) > t 4. 4( - b) < (b + 4) 5 4. 5x + x 44. a + a 5 4 45. 5 x < x 6 46. x > 4 47. x 4 48. x 5 49. + (x + 5) > x + 5(x + ) + 50. - 5t < 8

4- PROBLEM SOLUTIONS. x > 6. x < 50. x 50 4. x 5. y < -4 6. x < -7 7. r 8. x 9 9. x < - 0. t > -. m. x 405. x < 6 4. x > 5 5. x 6 6. b 5 7. x > 6 8. x < 9. k 0. x. x > 5. r > 7 6. s > 0. x 70 9 6. x > 7. t 4. s - 8. c 9. v < 9 0. w 5 7. k > 0 4. t > 5. x 50 9 5. x 4 50. t 96 7 6. r 7. s > 4 8. y > 4 9. x 40. a 9 8 4. t > 5 4. b > -4 4. x 6 44. a 5 7 45. x > 5 46. x > 6 47. x 9 8 48. x 0 49. x < 7 4 50. t > -

4- QUADRATIC INEQUALITIES Definition Procedure Quadratic inequalities in one variable are inequalities which can be written in one of the following forms: ax + bx + c > 0, ax + bx + c < 0, ax + bx + c 0 or ax + bx + c 0 where a, b and c are real numbers. Solving Quadratic Inequalities. Move all terms to one side.. Simplify and factor the quadratic expression.. Find the roots of the corresponding quadratic equation. 4. Use the roots to divide the number line into regions. 5. Test each region using theinequality. Example Solve the inequality, x > x +. x > x + x x > 0 (x - )(x + ) > 0 The corresponding equation is (x - )(x + ) = 0 so x - = 0 or x + = 0 x = x = - I II III -7-6 -5-4 - - - 0 4 5 6 7 Now we test one point in each region. Region Test Point Inequality Status I x = - (x - )(x + ) = (- - )(- + ) = 4 > 0 True II x = 0 (x - )(x + ) = (0 - )(0 + ) = - > 0 False III x = (x - )(x + ) = ( - )( + ) = 4 > 0 True -7-6 -5-4 - - - 0 4 5 6 7 So the solution to this inequality is x < - or x >.

Example Solve the inequality, (x + ) (x + 7). (x + ) (x + 7) x + 6x + 9 x +4 x + 6x 5 0 (x 6x + 5) 0 (x 6x + 5) 0 x 6x + 5 0 (x - )(x - 5) 0 The corresponding equation is (x - )(x - 5) = 0 so x - = 0 or x - 5 = 0 x = x = 5 I II III -7-6 -5-4 - - - 0 4 5 6 7 Now we check one point in each region. Region Test Point Inequality Status I x = 0 (x - )(x - 5) = (0 - )(0-5) = 5 < 0 False II x = (x - )(x - 5) = ( - )( - 5) = - < 0 True III x = 6 (x - )(x - 5) = (6 - )(6-5) = 5 < 0 False -7-6 -5-4 - - - 0 4 5 6 7 So the solution to this inequality is x 5.

4- PROBLEM SET Solve each quadratic inequality, and graph the solution on a number line.. y 7y + 70 < 0. x + 9x + > 7. x(x +) > 5x 4. a + a + < ( a + ) 5. x 5x 6. 0 9y y 7. b(b + ) 8. a 4(a ) 9. y 7y + 70 < 0 0. x + 9x + > 7. x(x +) > 5x. a + 5 <0a. d + 5d 4. a + a + ( a + ) 5. 0 9y y 6. x 5x 7. c(c + 4) < + (9 + c) 8. a(a + 6) > 5 a(a + ) 9. b(b + ) > 0. a < 4(a ). (x + ) 6(x +5). x + 7 9x. 7x 4(+ x) 4. x + 7x 5. 8 < 4(x x ) 6. x x > 0 7. k + k > 0 8. t + t < 0 9. 4x + 8 x 0. x 4(x 5). x + 4 x x. 0 x x. 4 < x x 4. 6(x + ) > 5. 6x x > 8 6. 0a < a 7. 8x ( x ) 8. y 5 9. t +8 t 40. x(x + ) x(x + 5) 4. x < 8 4. x + x > 4. t > 9t +8 44. 4x 9x + < 0

4- PROBLEM SOLUTIONS. 7 < y < 0. x < -5 or x > -4. x < -4 or x > 8 4. -4 < a < - 5. x 6. y or y 5 7. b - or b - 8. a 6 9. 7 < y < 0 0. x < -5 or x > -. x < -4 or x > 8. no solution. 4 d 4. x -4 or x - 5. y or y 5 6. x 7. -6 < c < 5 8. a < -5 or a > 9. b - or b - 0. < a < 6. -9 x 9. x < or x 7. x 7 or x 4. x 5. - < x < 6. x < - or x > 7. k < - or k > 8. - < t < 9. x 8 0. all real numbers 4. - < x < 4. x -5 or x. < x < 4 4. no real solutions 5. < x < 4 6. 4 < x < 5 7. x or x 8. x -5 or x 5 9. t or t 9 40. 0 x 4. < x < 4. x < 57 or x > + 57 4. t < or t > 6 44. 4 < x <

4- ABSOLUTE VALUE INEQUALITIES Theorem If u < c where u is a variable expression and c is non-negative, then c < u < c. Example Solve the inequality, x 6. If x 6 then 6 < x < 6. -7-6 -5-4 - - - 0 4 5 6 7 Example Solve the inequality, x + 4. If x + 4 then 4 x + 4. 4 x + 4 4 x + 4 6 x -7-6 -5-4 - - - 0 4 5 6 7 Example Solve the inequality, x + > 8. x + > -8 x + > -8 - x > -9 x < 9 x < If x < then < x <. < x < + < x + < + < x < 5-7 -6-5 -4 - - - 0 4 5 6 7

Theorem If u > c where u is a variable expression and c is non-negative, then u > c or u < -c. Example 4 Solve the inequality, x > 5. If x > 5 then x > 5 or x < -5. -7-6 -5-4 - - - 0 4 5 6 7 Example 5 Solve the inequality, x 5. If x 5 then x - 5 or x - -5. x - 5 x - -5 x - + 5 + x - + -5 + x 6 x -4-7 -6-5 -4 - - - 0 4 5 6 7 Example 6 Solve the inequality, x +0 <. x +0 < x +0 0 < - 0 x < -8 x > 8 x > 4 If x > 4 then x - > 4 or x - < -4. x - > 4 x - < -4 x - + > 4 + x - + < -4 + x > 5 x < - -7-6 -5-4 - - - 0 4 5 6 7

4- PROBLEM SET Solve each absolute value inequality, and graph the solution on a number line.. x > 4. x + < 5. v 5 4. x + 5 5. y < 5 6. y 5 > 9 7. t 8. t 9. p > 4 0. 4 b <. x 4. 5x + 0. w < 4 4. 4 x > 0 5. x + 8 6. a a 7. 4x > 0 8. x + <5 9. x + 0. x + 5. x <. 7z + >. 4 a 4 4. x + 4 6 5. + q > 5 6. x + + 6 < 0 7. x 4 8. 54x + 7 5 9. a 7 +8 <0 0. x + 9 >. t 4. 54x + 7 5. 4x 7 5 > 5 4. x + 4 < 4 5. n + 4 6. 5 4x 7 5 5 7. x + < 8. x + 5 > 9. x 5 40. 6y 4 4. x > 4. < x 4. y 9 5 44. 4 5 r + 9 45. h 4 7 >

4- PROBLEM SOLUTIONS. x > 4 or x < -4. - < x <. v 5 6 or v 5 6 4. - x 5. -5 < y < 5 6. y > 4 or y < -4 7. -4 t 4 8. t 4 or t -4 9. p > 4 or p < 4 0. < b < 4. x 6 or x. 5 x 9 5. - < w < 4. x > 5 or x < -5 5. 7 x 9 6. a or a 7. x > 5 or x < -5 8. < x < 9 9. x 0 or x -4 0. x 7 or x -. x < 0 or x >. z > 7 or z < 7. 4 a 5 4. x 8 or x -0 5. q > 4 5 or q < 5 6. 5 < x < 7. x 5 or x - 8. x 9. < a < 0. x > or x < -0. -8 t 8. x. x > 7 4 or x < 7 4 4. < x < 0 5. n 7 6 or n 6. x = 7 4 7. no solution 8. -4 < x < 8 9. x 7 40. y 7 6 or y 6 4. x < 0 or x > 4 4. 4 < x < 0 4. y 4 or y 4 44. 5 < r < 65 6 45. n > 7 or n <

5- COORDINATE GEOMETRY Definition A coordinate plane consists of a plane divided into four quadrants (I, II, III and IV) s by two perpendicular number lines called the x-axis and the y-axis. The point where the axes intersect is called the origin. y-axis II I origin III x-axis IV Example Plot each of the following points on a coordinate plane: A(, ), B(-4, ), C(-, -), D(4, -), E(0, -), F(, 0), O(0, 0) Plotting the points produces the plot below. B A O F C E D The results above illustrate the fact that the general location of a point can be determined by the signs of its coordinates. These results are summarized below. Summary x-coordinate y-coordinate Location + + Quadrant I - + Quadrant II - - Quadrant III + - Quadrant IV Any 0 x-axis 0 Any y-axis 0 0 Origin

Formula Formula The Midpoint Formula The coordinates of the midpoint between two points in a plane (x,y ) and (x, y ) are given by: x M + x, y + y The Distance Formula The distance between two points in a plane (x,y ) and (x, y ) is given by: d = (x x ) +(y y ) Example Consider two points A(5, -) and B(-, ). a. Find the midpoint for these two points. b. Find the distance between these two points. a. Let point A be the first point and point B be the second point. M x + x, y + y = M + 5, +( ) = M 4, = M(, -) b. Let point A be the first point and point B be the second point. d = (x x ) + (y y ) = [( ) 5] +[ ( )] = ( 6) + (4) = 6 +6 = 5 = 4 = Formula Formula In three dimensions, the midpoint formula and distance formula are slightly different. The Midpoint Formula in Three Dimensions The coordinates of the midpoint between two points in space (x,y,z ) and (x, y, z ) are given by: M x + x, y + y, z + z The Distance Formula in Three Dimensions The distance between two points in space (x,y,z ) and (x, y, z ) is given by: d = (x x ) +(y y ) + (z z )

5- PROBLEM SET Use the diagram below to answer the questions. A G 4 B C -4-0 4 E H F - -4 D Write the coordinates for each point.. A. B. C 4. D 5. E 6. F 7. G 8. H In which quadrant or on which axis is each point? 9. A 0. B. C. D. E 4. F 5. (500, -0) 6. (-5, -50) Find the midpoint of the segment joining each pair of points. 7. E and H 8. C and G 9. A and C 0. F and D. C and D. A and D. A and E 4. F and H 5. (-69, 8) and (5, -7) 6. (, -4) and (5, 0) 7. (6, -, 9) and (8, 7, ) 8. (60, -4, -0) and (5,, -7) Find the distance between each pair of points. Simplify if possible. 9. E and H 0. C and G. A and C. F and D. C and D 4. A and D 5. A and E 6. F and H 7. (-69, 8) and (5, -7) 8. (, -4) and (5, 0) 9. (6, -, 9) and (8, 7, ) 40. (60, -4, -0) and (5,, -7)

5- PROBLEM SOLUTIONS. (-4, ). (4, 4). (, 0) 4. (, -) 5. (-4, -) 6. (0, -) 7. (-, 4) 8. (-, -4) 9. Quadrant II 0. Quadrant I. x-axis. Quadrant IV. Quadrant III 4. y-axis 5. Quadrant IV 6. Quadrant III 7. (-, -) 8. (, ) 9., 0.,.,.,0. 4, 4., 7 5. 7, 6. (9, ) 7. ( 7,, ) 8.,, 67 9. 0. 4. 58.. 4. 85 5. 5 6. 5 7. 9 8. 85 9. 0 40. 55

5- LINEAR RELATIONS Definition The slope of a line measures the direction that a line travels. It can be calculated using the formula m = y y x x where (x, y ) and (x, y ) are two points on the line. Lines with positive slope increase from left to right, while lines with negative slope decrease from left to right. Horizontal lines have slope zero while the slope of vertical lines is undefined. Definition The y-intercept of a line is the y coordinate of the point where the line crosses the y- axis. There are two primary forms in which linear relations can be expressed. Definition Definition The slope-intercept form for the equation of a line is y = mx + b where m is the slope of the line and b is the y-intercept. The standard form for the equation of a line is Ax +By = C where A, B and C are real numbers. Example Find the slope and y-intercept for the line x + y = 9. Solving for y changes the line into slope-intercept form. x + y = 9 y = -x + 9 y = x + Thus the slope is and the y-intercept is. Example Find the equation (in slope-intercept form) of the line that passes through the points (-, ) and (6, 7). The slope of the line is m = y y x x = 7 6 ( ) = 4 8 = So the equation becomes y = x + b. Substituting the point (6, 7) into the equation produces: 7 = 6 + b Therefore the line has the equation: y = x + 4. 7 = + b 7 - = + b - 4 = b

Property Two lines are parallel if they have the same slope. That is m = m where m and m are the slopes of the two lines. Example Find the equation (in slope-intercept form) of the line that is parallel to the line y = x and passes through the point (-, ). The slope of the original line is so the slope of the new line must also be. So the equation for the new line is y = x + b. Substituting the point (-, ) into the equation produces: = ( ) + b = - + b + = - + b + = b Therefore the line has the equation: y = x +. Property Two lines are perpendicular if their slopes are negative reciprocals of each other. That is, m = m where m and m are the slopes of the two lines. Example 4 Find the equation (in slope-intercept form) of the line that is perpendicular to the line y = x and passes through the point (-, ). The slope of the original line is so the slope of the new line must be -. So the equation for the new line is y = x + b. Substituting the point (-, ) into the equation produces: = ( ) + b = 9 + b - 9 = 9 + b - 9-8 = b Therefore the line has the equation: y = x 8.

5- PROBLEM SET Find the slope and y-intercept for each line below. 4x - y = 5. y - x = 4. y = 4 x + 7 4. y = x + 5. y + x = - 6. x + y = 4 7. 5x - y = 0 8. = y + x 9. x + 5y = 0 5 0. x + y = 6. y = 5. x 5 6 y + = 0 Write an equation in slope-intercept form for each of the following lines.. The line with a slope of and a y-intercept of. 4. The line with a slope of - and a y-intercept of 4. 5. The line with a slope of 5 which passes through the point (, ). 6. The line with a slope of - which passes through the point (4, -). 7. The line which passes through the points (-, -5) and (0, ). 8. The line which passes through the points (5, -) and (-, 4). 9. The line which is parallel to y = -x + 7 and passes through the point (, 0). 0. The line which is parallel to y = x + and passes through the point (, ).. The line which is perpendicular to x - y = - and passes through the point (, 0).. The line which is perpendicular to y = x and passes through the point (, 5).. The line which is parallel to y = 5 and passes through the point (4, -). 4. The line which is perpendicular to y = - and passes through the point (-, -). 5. The line which is perpendicular to x = 4 and passes through the point (-4, ). 6. The line which is parallel to x = - and passes through the point (-5, -). 7. A line segment has endpoints (-5, 4) and (, -). Find the equation of the line which is perpendicular to this segment and which passes through the midpoint of the line segment.

5- PROBLEM SOLUTIONS. m = ; b = 5. m = ; b = 4. m = 4 ; b = 7 4. m = ; b = 4 5. m = -; b = - 6. m = ; b = 7. m = 5 ; b = 0 8. m = 0 ; b = 9. m = 5 ; b = 0. m = ; b =. m = 0; b = 5. m = 8 5 ; b = 5. y = x + 4. y = -x + 4 5. y = 5x - 6. y = -x + 7 7. y = x + 8. y = 4 x + 7 4 9. y = -x + 4 0. y = x +. y = x +. y = -x + 7. y = - 4. x = - 5. y = 6. x = -5 7. y = x -

5- SYSTEMS OF LINEAR RELATIONS Procedure Solving Two Linear Equations by Substitution Step : Choose either variable and solve one of the equations for that variable. Step : Substitute the result into the other equation. This will produce an equation with one variable. Step : Solve the equation for the remaining variable. Step 4: To find the other variable, substitute the result from Step into the equation developed in Step. Step 5: Write the solution as an ordered pair and check the answer. Example x + y = 8 Solve the system of linear equations 4x + y = 4 by the substitution method. Step : Choosing the first equation, solve for x. x + y = 8 x = 8 - y x = 8 y Step : Now substitute this expression for x into the second equation. 4x + y = 4 4 8 y + y = 4 Step : Now solve this single variable equation for y. (8 - y) + y = 4 6-6y + y = 4 6 - y = 4 -y = - y = 4 Step 4: Finally substitute the result y = 4 into x = 8 y from Step. x = 8 y = 8 (4) = 8 = 4 = - Step 5: The solution to the system is (-, 4). Check this solution in both equations. x + y? 8 4x + y? 4 (-) + (4)? 8 4(-) + (4)? 4-4 +? 8-8 +? 4 8 8 4 = 4

Procedure Solving Two Linear Equations by Elimination Step : Write both equations in standard form: Ax + By = C. Step : Multiply one or both of the equations by appropriate numbers so that the coefficients of either x or y are the same. Step : Add or subtract the two equations to produce an equation with only one variable. Step 4: Solve the equation for the remaining variable. Step 5: To find the other variable, substitute the result from Step 4 into either of the two original equations. Step 6: Write the solution as an ordered pair and check the answer. Example Solve the system of linear equations x y = 0 x + y + = 0 by the elimination method. x y = Step : Write both equations in standard form: x + y = Step : Multiply the first equation by to make the coefficients of y equal. x - y = (x - y) = () 4x - y = 4 Step : Next, add the two equations together to eliminate the y variable. 4x - y = 4 + x + y = - 7x + 0y = Step 4: Now solve this single variable equation for x. 7x = x = Step 5: Finally, substitute the result x = into x + y + = 0 we can solve for y. x + y + = 0 () + y + = 0 y + = 0 y = - y = -6 Step 6: The solution to the system is (, -6). Check this solution in both equations. x - y -? 0 x + y +? 0 () - (-6) -? 0 () + (-6) +? 0 6 + 6 -? 0 9 - +? 0 0 = 0 0 = 0

5- PROBLEM SET Solve each system of equations using substitution.. x + y = x = y. 9x + 5y = 8 x y = 4. 5x + y = 9 7 x = y 4. x 5y = x y = 5. x = y 9 = x + 7y 6. y x = x + 5y = 7 7. 5x 8y = 5x 9 = y 8. x + y = x + 4y = 9. y = x + = y + x Solve each system of equations using elimination. 0. x + y = x y = 7. x y = 5 x + y = 5. x + y = x y = 4. x + 4 = y 5x y = 4. x + y = 4 x y = 5. x + 7y = 4 x + = y 6. 5x + y = 40 x + y = 4 7. 4x = y + 9 8 = x y 8. 6x = y + 5 y = 9x + 4 Solve each system of equations using any method. 9. x y = 7 x y = 9 0. x y = 8 x + y =7. x y = 8 x y =. 4x y = 5 0x + y =. x + 9y = x 6y = 0 4. x y = 5x +0y = 5. x y = 7 4x + y = 0 6. 0.4x 0.y =.4 0.6x + 0.9y =.4 7. x +.5y = 0.5 0.8x 0.5y = 0.9 8. y = x = x + y 9. x + y = 0 x + y = 0 0. 7x 5y 4 = x + 4 = y 4. x y = y + x = 0. x 4 + 5y = 8 x 5 = y. y x + y 4 = x + 4 = 0

5- PROBLEM SOLUTIONS. (, ). (-, -). (4, ) 4. (, ) 5. (-, 4) 6. (, ) 7. (, ) 8. (, -) 9. (-, ) 0. (9, ). (6, 7). 0, 8. (, 8) 4. (4, ) 5. (-5, ) 6. (, -0) 7. (, -5) 8. (-, -) 9. (, -) 0. (5, ). (, -5).,., 4. 5, 7 0 5., 6. (7, -) 7. (, ) 8. (, -) 9. (, ) 0. (-, 4). (-, ). (4, ). (5, )

5-4 FUNCTIONS Definition A function is a rule that assigns a unique real number to each number in a specified set of real numbers. Functions are expressed in the form f(x) = u where u is a variable expression. f(x) does not indicate that f is multiplied times x but rather that the function f should be evaluated at the value x. Example Evaluate the function f (x) = x +x 7 at the indicated values. a. x = b. x = - c. x = t d. x = b + a. f() = () +() 7 = (4) + - 7 = 8 + 5 = b. f(-) = ( ) +( ) 7 = () - - 7 = - 8 = -6 c. f(t) = (t) +(t) 7 = t +t 7 d. f(b + ) = (b +) +(b +) 7 = (b + b +) +b + 7 = b + 4b + +b + 4 = b +5b + 6 Theorem The Vertical Line Test If a vertical line drawn anywhere on the graph of a relation (in x and y) will intersect the graph at no more than one point, then the relation is a function of x. Example Use the vertical line test to determine which of the following relations are functions of x. a. b. c. d. a. This is a function since any vertical line will intersect the graph at one point. b. This is not a function many vertical lines will intersect the graph at two points. c. This is a function since any vertical line will intersect the graph at one point. d. This is a function since any vertical line will intersect the graph at one point.

Functions can be translated (moved) by using the following rules. Properties Function Description g(x) = f(x) + k g(x) is f(x) translated k units up g(x) = f(x) - k g(x) is f(x) translated k units down g(x) = f(x - h) g(x) is f(x) translated h units to the right g(x) = f(x + h) g(x) is f(x) translated h units to the left Example Use the graph of f(x) below to graph each of the following functions. f(x) - -4 a. g(x) = f(x) + b. h(x) = f(x) - c. u(x) = f(x - ) d. v(x) = f(x + ) a. g(x) is f(x) translated units up. b. h(x) is f(x) translated unit down. g(x) h(x) - - -5 c. u(x) is f(x) translated right units. d. v(x) is f(x) translated left units. v(x) u(x) -4-4

5-4 PROBLEM SET Evaluate each function at the indicated values.. f(x) = x + 5; x = -. g(x) = x ; x =. h(t) = t +; t = - 4. f (a) = a + a 7 ; a = 5. g(x) = 5 x + ; x = 4 6. h(x) = x + ; x = 4 7. u(r, s) = r + ; r = 5, s = - 8. g(x, y) = xy y + y ; x = -, y = 4 s Evaluate the function f(x) at the indicated values: f (x) = x + x 9. f(0) 0. f(). f(a). f(x + ) Evaluate the function g(x) at the indicated values: g(x) = x + x. g(0) 4. g() 5. g(a) 6. g(x + ) Use the function f(x) to determine g(x) in each case: f (x) = x x. 7. g(x) = f(x) + 5 8. g(x) = f(x) - 9. g(x) = f(x - ) 0. g(x) = f(x + ). g(x) = f(x - ) +. g(x) = f(x + ) - Use the graph of the function f(x) to draw the graph of g(x) in each case. f(x) (-5, ) (-, -) (4, -) (, -). g(x) = f(x) + 4. g(x) = f(x) - 5. g(x) = f(x + ) 6. g(x) = f(x - ) 7. g(x) = f(x - ) + 8. g(x) = f(x + ) -

5-4 PROBLEM SOLUTIONS. -.. 6 4. 5. 6. 7 7. 8. -0 9. 0. 8. a + a. x + 4 x. - 4. 9 5. a + a 6. x + 5x + 7. x x + 4 8. x x 9. x 9x + 9 0. x +x + 9. x 7x + 5. x +x + 6. 4. (-5, 4) (4, ) (-, 0) (, 0) (-5, -) (4, -) (-, -5) (, -5) 5. 6. (-6, ) (-, ) (, -) (6, -) (-, -) (, -) (, -) (4, -) 7. 8. (-4, ) (5, ) (-7, 0) (0, -) (, -) (, -) (-, -4) (0, -4)

6- PERIMETER AND CIRCUMFERENCE Definition Perimeter is a measure of the distance around a figure. For a polygon, the perimeter is the sum of the lengths of its sides. For a circle, the measure of the distance around a circle is called the circumference. Definition Formula The circumference of a circle is the perimeter of the circle. The circumference of a circle is given by the equation: C = πr where r is the radius of the circle (the distance from the center of the circle to its edge). The diameter of a circle is twice its radius. Example For the diagram below, find the perimeter of the square and the circumference of the circle. diameter = 6 cm Since the circle is inscribed inside the square, the diameter of circle is equal in length to the side of the square. Since the square has four equal sides, it's perimeter is: 6 cm + 6 cm + 6 cm + 6 cm = 4 cm Since the diameter of the circle is 6 cm its circumference is given by: C = πr = π 6 = π Example Example A certain square has the same perimeter as a 0 by foot rectangle. How long is a side of the square? The length of the rectangle is 0 ft and its width is feet so its perimeter is given by: P = l + w = 0 + = 4 Since the square has four equal side lengths, each of its sides is one-fourth of 4 ft. Therefore the square has sides which are 6 ft in length. The width of a rectangle is twice its length. If its perimeter is 7 cm, find the length and width. If the length is given by l, then the width is w = l. Therefore P = l + w = l + (l) = l + 4l = 6l Thus 6l = 7 and l = 4.5. So the length is 4.5 cm and the width is 9 cm.