Chapter w. 5. a. 3(2x 7) = 3(2x) + 3( 7) = 6x a. 4(9x + 1) + 6 = 36x = 36x Subtract 7x 1 from 2x + 3 translates to

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1 Chapter Section. Practice Exercises. a. The numerical coefficient of t is, since t is t. b. The numerical coefficient of x is. c. The numerical coefficient of since w means w. d. The numerical coefficient of w is, x is. e. The numerical coefficient of b is, since b is b.. a. xy and yx are like terms, since xy yx by the commutative property. b. q and q are unlike terms, since the exponents on q are not the same. c. d.. a. ab, ab, and ab are like terms, since each variable and its exponent match. y and y are like terms, since the exponents on y are the same. x + x (+ ) x x b. y + y y + y ( + )y y. a. y+ 8y + (+ 8) y+ ( + ) y b. x x x x+ ( ) ( ) x + ( ) x c. t t t t t t d. y+.y+ 0 + ( +.) y+ (0 + ).y + e. z z These two terms cannot be combined because they are unlike terms.. a. (x ) (x) + ( ) x b. (x z ) ( x) + ( )( z) + ( )( ) x+ 0z+ c. (x y+ z ) (x y+ z ) ( x) ( y) ( z) ( ) x+ y z+. a. (x + ) + x + + x + 0 b. (x ) ( x) x+ + x x + c. 8 (x + ) 8 0x 0x. Subtract x from x + translates to (x+ ) (x ) x+ x+ x+ c. x x + 8x x+ ( + 8) x x+ x 8. a. Three added to double a number + x Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

2 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra b. the sum of and a number subtracted from six ( + x) + x ( + x) + x x c. d. two times the sum of and a number increased by ( + x) + ( + x) + + x + x + 0 a number added to half the number added to times the number x + x + x x+ x+ x x Vocabulary and Readiness Check. y 0y + is called an expression while y, 0y, and are each called a term.. To simplify x + x, we combine like terms.. The term y has an understood numerical coefficient of.. The terms z and y are unlike terms and the terms z and z are like terms.. For the term, xy the number is the numerical coefficient.. (x y) equals x y by the distributive property.. The numerical coefficient of y is. 8. The numerical coefficient of x is.. The numerical coefficient of x is. 0. The numerical coefficient of y is.. The numerical coefficient of y is. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

3 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. The numerical coefficient of. y and y are like terms. z is.. (z + ) z + z (x z ) ( x) ( )( z) ( )() 8x+ z+. x y and xy are unlike terms.. z and. ba and Exercise Set. z are unlike terms. 8 ab are like terms.. x + x ( + )x x. c c + c ( + )c c. g+ g g g+ ( ) g g 8. a+ a a a+ a a (+ ) a a 0. 8p+ 8p (8p 8 p) + ( ) (8 8) p + ( ) 0p..y 0. y+ 0..y y (. ) y 0..y 0.. 8h+ h + h h 8h+ h+ h h (8 + + ) h h. 8x + x x (8+ ) x x 8. 0.y. + y 0..y 0.y + y.y. 0. (0. +.)y.0.y 0. Answers may vary. (r ) (r) () r. (y + ) (y) + ( )() y 0. (y + z ) y z +. (x ) ( x+ ) 8x x x. y added to y+ (y ) + ( y+ ) y+ y + y +. + x minus x ( + x) (x ) + x x+ + + x x x 8. m minus m (m ) ( m ) m m+ m m + m 0. c 8 c c c 8 ( )c 8 c 8. y + y 0y y+ y 0y ( + 0) y 8y. (x + ) x ( x) + ( )() x x x x x x. (x ) ( x ) x x+ x y ( y+ ) 8y y y 0. c ( c) c + c c. (8 y) ( + y) 8 y y 8y+..8w w.8w.8w Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

4 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra. (y+ ) + (y ) y+ + y y+ y+ 0 0 y y (x ) 8 + x 8 + x 0. 0.( k + 8) 0.k 0.k+. 0.k 0.k+.. (m + n) m n. (x + ) ( x+ ) 0x x+ x 8 0x x x 0x+ 8 0x ( x ) ( x ) x x+ x The difference divided of a number by and x ( x ) 0. 8 more than triple a number 8 + x.. eleven increased two-thirds of by a number + x times a times a subtract number number and 0 x (x+ 0) x (x + 0) x x 0 x 0 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

5 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving the difference Six times of a number and ( x ) (x ) x 0 half a the product of minus number the number and 8 x 8 8. x x x the product of a number less 0 and 0 0x 0 twice a added added times a added number to to number to x + + x + x + ( ) + x + ( ) x gh h 0( ) ( ) 0 x x + ( ) ( ) + + x x x ( ) ( ) ( ) (x ) + (x+ ) + x + x+ x + The perimeter is (x + ) centimeters.. cylinders cubes cubes + cubes cubes cubes cubes: Not balanced. cylinder cone + cube cubes cube + cube cubes cubes: Balanced. Answers may vary Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

6 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra 8. x+ 0( x) + (0x ) x+ 0x+ 0x 8x The total value is (8x ) m p m p m p m p m p y (xy y ) 8xy y xy + y 8xy y xy (cd 8 c) c cd cd+ 8c c cd cd+ c Section. Practice Exercises. x + x + x 8 Check: x + 8+ The solution is 8.. y 0.. y y.8 Check: y The solution is.8.. 8x x + x+ x+ x+ x x+ x x x x + x + x 0 Check: 8x x + x+ x+ 8( 0) ( 0) ( 0) ( 0) + The solution is 0.. (a ) (a+ ) ( a) + ( ) a 8a a a + a + 8 a Check by replacing a with 8 in the original equation.. x x x x 0 x 0 Check: x 0 The solution is 0.. 8x 8x 8 8 x Check: 8x 8( ) The solution is. x. x x x Check: The solution is. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

7 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving 8. b b 8+ b + b + b b b + b b b b b Check by replacing b with in the original equation. The solution is.. a. The other number is. b. The other number is x. c. The other piece has length ( x) feet. 0. Let x first integer. x + second even integer. x + third even integer. x + (x + ) + (x + ) x + Vocabulary and Readiness Check. The difference between an equation and an expression is that an equation contains an equal sign, whereas an expression does not.. Equivalent equations are equations that have the same solution.. A value of the variable that makes the equation a true statement is called a solution of the equation.. The process of finding the solution of an equation is called solving the equation for the variable.. By the addition property of equality, x and x are equivalent equations.. The equations x and x are equivalent equations. The statement is true.. By the multiplication property of equality, y and y are equivalent equations. z z 8. The equations 0 and 0 are not equivalent equations. The statement is false. x 0. The equations x 0 and are not equivalent equations. The statement is false. 0. By the multiplication property of equality, x and x are equivalent equations.. a a. c c. b 0 0 b. t t Exercise Set.. x + x + x Check: x + + The solution is.. y y + + y 0 Check: y 0 The solution is z z 8 8 z Check: 8+ z ( ) The solution is. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

8 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra 8. t t. Check: t The solution is.. 0. x x x x x x x Check: x x ( ) 0 0 The solution is.. x+. 0x x x+. 0x x. x Check: x+. 0x (.) +. 0(.). +. The solution is... 8x x 8x 8x x 8x x Check: 8x x 8( ) ( ) The solution is.. y+ y+ y y+ y y+ y + y + y 0 The solution is c+ 8 c 8+ c c c c c c c c c c 0 The solution is n+ n + n n + n n n + n n n The solution is.. 0 8(y ) y+ 0 0 y y+ 0 0 y 0 + y + y The solution is.. x x x The solution is.. x 0 x 0 x 0 The solution is y 8 y 8 y 8 The solution is y+ y y y y The solution is... n n ( ) n 0 The solution is 0. v 8 8 v 8 8 v The solution is. 8 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

9 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving d. d () d 0 The solution is 0. f 8. 0 f (0) f 0 The solution is Answers may vary. x x + + x x x Check: x () The solution is.. x + x + x 8 x 8 Check: x + (8) The solution is 8.. 8t + 8t + 8t 0 8t t 0 Check: 8t + 8(0) + 0+ The solution is y + 0y + 0y 0 0y y Check: 0y The solution is p + 0.p 0. p 0 0. p p 0 Check: + 0.p The solution is n 8 n + + n n n n 8 Check: n 8 ( ) The solution is. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

10 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra b. b + + b b ( ) b b Check: The solution is.. j + j + j j j Check: j The solution is. 8. a+ + a 0 a 0 0 a a 0 a 0 a Check: a+ + a The solution is. 0. x x 0 0x + 0 0x + 0 0x 0x 0 0 x 0 Check: x x The solution is. 0. x x x Check: x ( ) The solution is.. 0.x 0.x 0.x + 0.x + 0.x 0.x x Check: 0.x 0.x 0.( 0) 0.( 0) 0 + The solution is 0. 0 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

11 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. t t + t t t t t+ t t+ t t t t Check: t t + t t The solution is x 0.x 0.x 0.x x 0.x x 0 Check: 0.x 0.x 0.( 0) 0.( 0) 0 + The solution is y + y + y + y 8 y 8 y Check: y + ( ) The solution is.. b 8 b 0b b b 8 b b b 8 b b 8 b 8 b b Check: b 8 b 0b b ( ) 8 ( ) 0( ) ( ) The solution is.. (x + ) + x 0x + x x + x + x Check: (x+ ) + x ( + ) + (8+ ) + () + + The solution is.. The other number is y. 8. The length of the other piece is ( x) feet. 80. The complement of the angle x is (0 x). 8. The length of the computer desk is m + feet. 8. The length of I-0 is (m + 8.) miles. 8. Susan received (n + 0,88) votes. 88. The weight of the Hoba West meteorite is y kilograms. 0. Sum first integer + second integer + third integer + fourth integer. Sum x+ ( x+ ) + ( x+ ) + ( x+ ) x+ x+ + x+ + x+ x + Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

12 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra. Sum 0 + second integer. Sum 0 + ( x + ) 0 + x + x +. Let x be an odd integer. Then x + is the next odd integer. x + (x + ) + x + (x + ) x +. y+ y ( y+ ) y+ y y 8y 8. (a ) + a a + + a a 00. ( ) ( )( )( )( ) ( ) > 0. ( ) ( ) ( )( )( ) 0. 0 (x + x + x) 0 (x) 0 x The fourth angle is (0 x). 0. Answers may vary 08. a + a + + ( ) + ( ) a The answer is. 0. Answers may vary. Answers may vary. x x. x. x. Each dose should be. milliliters. 8. Check a.. ( a+.) a+. (. +.) (.) +. (0.). +.. Not a solution 0..y.8.y.8.. y.. 0.0y y y y y. Section. Practice Exercises. (a ) + a 8a 8+ a 8a a 8a a a a a a + + a a a Check: (a ) + a [() ] + () ( ) + () + + The solution is or the solution set is {}.. ( x ) x x x x x x x x x x Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

13 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving Check: ( x ) x 8 The solution is.. 0.x+ 0.0( x+ ) 0.0() 00[0.x+ 0.0( x+ )] 00[0.0()] x+ ( x+ ) () x+ x+ x + x + x 0 x 0 x 0 To check, replace x with 0 in the original equation. The solution is 0... x x x x x () x () x 0 0x x 0 x 0x x 0 x 0 + x + x Check: x x 0 The solution is. ( y + ) y ( y + ) (y ) ( y+ ) (y ) y+ y y+ y y y y + y + y y y To check, replace y with in the original equation. The solution is.. ( x+ ) x ( x+ ) + x x+ x x+ + x x+ x+ x+ x x+ x There is no solution.. x 8 ( x ) + x x 8 x 8 + x x 8 x 8 x x x x x x x x 0 0 The solution is all real numbers. Calculator Explorations. Solution ( ). Solution ( ). Not a solution (. 0.). Not a solution (. 0.). Solution (,0,0). Solution ( ) Vocabulary and Readiness Check. x is an equation.. x is an expression.. y + y + is an expression.. y y + is an equation. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

14 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra. x is an expression. x 8. x is an equation. x 8. 0.x + 0.x is an equation x y 0.x + is an expression. Exercise Set.. x+ (x+ ) x+ 8x x+ 8x x 8x x + 8x 8x + 8x x x x. x + x x + + x+ x + x x x + x x x x x. (x 0) x x+ 0 x x + 0+ x x+ x 0 0x 0 0x 0 0 x 0. ( n ) n + n n + + n 0 n 0 n 0. ( + b) b b b b b b b b b b + b + b b b. y 8 + y+ y 8 y+ y y 8 y y+ y 8 y y y y. n+ 8n 0 n+ 8n 0 n 8n n 8n 8n 8n n n n 8. ( x) + ( x) x+ x + x + x x x x 8. 8 x 8 x x 8 x x 8 x 8 x Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

15 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving 0. x x () x x + + x x x. 0.0x + 0.0(0).8 00[0.0x + 0.0(0)] 00(.8) 0x + (0) 80 0x x x 800 0x x 0.. ( y + ) y + ( ) y + [ ] y + ( y+ ) 0y+ 0 y+ 0y+ 0 y 0y+ 0y 0y+ 0 y + 0 y + 0 y y y x x+ x x+ 0x x+ 0x x x x+ x x + + x x x ( z 00) + 0.0z 0.0z 0 00[0.0( z 00) z] 00[0.0z 0] 0( z 00) + z 0z 0, 00 0z 8, z 0z 0, 00 z 8, 000 0z 0, 00 z 0z 8, 000 0z 0z 0, 00 z 8, 000 0, 00 z 8, , 000 0,00 + 8,000 z 00 z 00 z x+ (x+ ) x+ x+ x + x x+ x All real numbers are solutions. x x. x x x x x x x x 0 There is no solution.. ( x ) x+ 0 x 0 x+ 0 x x 0 x x There is no solution.. (y ) + 0y+ 0y+ + 0y+ 0y+ 0y+ 0y+ + 0y 0y+ + 0y All real numbers are solutions. ( w) 8. w ( w) ( ) w ( w) w 0 w w 0 w+ w w+ w 0 w Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

16 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra 0. (a ) a 0 + a a+ a 0+ a a+ 0+ a a a+ 0+ a a 0a + 0 0a + 0 0a 0a 0 0 a 0. x+ ( x ) 0( x ) + x+ x 0x 0 + x 0x x + 0x + x 0x x 0x 0x 0x x x x. ( x ) ( x+ ) ( x ) ( x+ ) ( x ) ( x+ ) x x+ x x x x+ x x + + x x x. 0.x. 0. 0(0.x.) 0(0.) x x + + x x x 8. (x ) + x+ x + x+ x+ x+ x x+ x x+ All real numbers are solutions. 0. (y+ ) ( + y) + 8 y+ 8 + y+ 8 y y y y 8 y + y y y + y y y y y. x+ x 8 8 x + 8 x 8 x+ x x + x x x x x x x x. x x x 0 x x 0 x x x 0 x 0 + x + 0 x 0 x x. ( + x) + x ( x ) 8+ x+ x x+ + x x+ + x x x x+ There is no solution. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

17 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving ( x+ ) ( x+ ) 00[ 0.0( x+ )] 00[ ( x+ )] (x+ ) ( x+ ) x x x x x+ x x+ x x 0 x + 0+ x x x 0. x x 8 x (x 8) x 0x x + x 0x + x x + x + x x x. n+ 0n 0 n+ 0n 0 n 0n n 0n 0n 0n n n n. 0.x 0. 0.x. 0(0.x 0.) 0(0.x.) x x x x x x x x + + x 0 x 0 x. 0.0(m+ ) 0.0( + m) [0.0(m+ )] 00[0.0( + m) 0.0] (m+ ) ( + m) m+ 0+ m m+ + m m m+ + m m All real numbers are solutions. 8. times a number x x minus twice a number 8 x the difference the quotient and of a number of and ( x ) x. x + (x ) x + x 8x The total length is (8x ) feet.. a. x+ x+ x + x x+ x There is no solution. b. Answers may vary c. Answers may vary 8. x+ x+ x + x x+ x There is no solution. The answer is b. 80. x x 0x 0x 0x 0x + 0x 0x + 0x All real numbers are solutions. The answer is a. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

18 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra 8. x+ x+ x + + x x+ + x x + x + 0 x 0 x 0 x The answer is c. 8. Answers may vary 8. a. Since the perimeter is the sum of the lengths of the sides, x + x + + x. b. x x + + x x x c. x + () + x () The lengths are x meters, x + meters and x meters. 88. Answers may vary ( x+ 0) 00( + x) 000x+ 0, x 000x + 0, x x 00x 00x + 0, x + 0, 000 0, , x 8, 00 00x 8, x 8. 0.x.8 0.0x.8 000(0.x.8) 000(0.0x.8) x 8 x 80 x x 8 x x 80 00x x x 0 00x x.0.. t t t(8 + t) t t 8t+ t t t t 8t+ t t t 8t t+ t 8t+ t 0 t 0 t 0 t y y+ 0 y( y ) y y+ 0 y y y y y+ 0 y y y y+ 0 y y+ y+ 0 y+ y y y y 0 The Bigger Picture. x (x ) + x x + x x+ x x x+ x x x + + x 8 x 8 8 x. + x ( x+ ) + x x+ + x x x+ x x x x 0 x 0 x 0 8 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

19 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving... x + x + () x + x + x x x x x ( x ) ( x ) x x+ 8 x + x + x y + 0 y 0 + 0() 0 () + y 0 + y 0 + y 0 y. + x ( x+ ) + x x+ + x x x+ x False This false statement indicates that there is no solution.. ( x ) + x ( x ) x 8+ x x x 8 x x 8 x x x x 8 x x x x 8. ( x+ ) x+ x+ x+ x+ x+ x + x x+ x True This true statement indicates that all real numbers are solutions of the equation. Integrated Review. x 0 x x. y + y + y. y 08 y 08 y. x 8 x 8 x. x + x + x 8 x 8 x. y y + + y y y. x () x x Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

20 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra 8. z 0 (0) z z r. r ( ) r 8 y y 8 8(8) 8 y. x x + 0 x + 0 x x x. y + y + y + y 8 y 8 y. x x x x x x There is no solution.. + 8y 8y + 8y 8y 8y 8y There is no solution.. a+ + a a 8a a+ a a a+ a a a a a. b 8 b 0b b b 8 b b b 8 b b 8 b 8 b b. 8. x x x y y 8 y. 0 n + 0 n + n n n 0. m + m + m m m 0 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

21 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. (c ) c+ c c+ c c+ c c c c+ c c + + c 8 c 8 c. (t+ ) 0 + t t+ 0 + t t + t t t + t t t t + + t t t.. ( z + ) z ( z + ) ( ) z z+ z z + z+ z+ z z + z + z z z ( w + ) w + ( ) w + ( ) w + w+ 8w+ w 8w+ 8w 8w+ w + w + w w w. (x ) x+ x+ x+ 0 x+ 0 x+ x+ 0 x+ x All real numbers are solutions.. (x ) x+ 8x+ 0x+ 8 0x+ 8 0x+ 0x+ 8 0x+ 0x All real numbers are solutions.. 0.0(t ) 0.0( t ) [0.0(t )] 00[0.0( t ) + 0.0] (t ) ( t ) + t t 8 + t t t t t t 8t 8t + + 8t 0 8t t ( m+ ) 0.0( m) [0.0( m+ )] 00[0.0( m) + 0.0] ( m+ ) ( m) + m+ 0 m+ m+ m m+ m+ m+ m m + m + m 8 m 8 8 m. ( y ). y ( y ) ( y) y y y y y y y y y Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

22 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra ( 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 x n+ n n+ ( ) n n+ n n n+ n n n n n (x ) x (x ) 0 0 x+ 0 x x+ 0 x x x+ 0 x 0 There is no solution.. (x ) x+ (x ) x+ x x+ x x x+ x There is no solution.. + (x ) (x ) + x x+ 8 x x+ 8 x + x x+ 8 + x 0x 8 0x x 0x 0 0 x. + (x ) (x+ ) + 0x 0 x 0x x 0x + x x + x x x + + x x x Section. Practice Exercises. Let x the number. x x+ x x x+ x x x + + x The number is. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

23 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x the number. x ( x ) x x x x x x x x + + x The number is.. Let x the length of short piece, then x the length of long piece. x+ x x x x x () The short piece is inches and the long piece is inches.. Let x number of Republicans, then x + number of Democrats. x+ x+ 0 x + 0 x + 0 x x x x There were Republican and 8 Democratic Governors.. x degree measure of first angle x degree measure of second angle x + degree measure of third angle x+ x+ ( x+ ) 80 x + 80 x + 80 x x x x () x The measures of the angles are,, and 80.. Let x the first even integer, then x + the second even integer, and x + the third even integer. x+ ( x+ ) + ( x+ ) x + x + x 8 x 8 x x x The integers are, 8, and 0. Vocabulary and Readiness Check. x; x. x; x +. x + ; (x + ). x ; (x ). 0 y; 0 y. 0 + y; Exercise Set. 0 + y. Let x the number. x x x x x x x x x x x x x The number is.. Let x the number. x+ ( ) x+ ( ) x x x + x + x x x x x x 0 x The number is 0. or (0 y) or ( 0 + y) Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

24 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra. Let x the number. [ x+ ( )] x x+ ( ) x x x x x x x x The number is. 8. Let x the number. ( x ) x x 8 x (x 8) x 8x x 8x x x x x x + + x x The number is. 0. Let x length of shorter piece, then x + length of longer piece. x+ (x+ ) x + x + x x x x + () + The shorter piece is feet and the longer piece is feet.. Let x area of Gobi Desert, then x area of Sahara Desert. x+ x,000,000 8x,000,000 8x,000, x 00, 000 x (00,000),00,000 The area of the Gobi Desert is 00,000 square miles, and the area of the Sahara Desert is,00,000 miles.. Let x number of television stations in China, then x + 0 number of television stations in Russia. x+ x+ 0 0, x + 0 0, x , 0 x 80 x 80 x 0 x China has 0 television stations and Russia has 0.. Let x the measure of angles B and C, and x the measure of A. x+ x+ x 80 x 80 x x x x x The angles are B, C, and A. 8. Three consecutive integers: Integer: x Next integers: x +, x + Sum of the second and third consecutive integers, simplified: (x + ) + (x + ) x + 0. Three consecutive odd integers: Odd integer: x Next integers: x +, x + Sum of the three consecutive odd integers, simplified: x + (x + ) + (x + ) x +. Four consecutive integers: Integer: x Next integers: x +, x +, x + Sum of the first and fourth consecutive integers, simplified: x + (x + ) x +. Three consecutive even integers: Even integer: x Next integers: x +, x + Sum of the three consecutive even integers, simplified: x + (x + ) + (x + ) x + Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

25 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x the number of one room and x + the number of the other. x+ x+ x + x + x x x x The room numbers are and Let x code for Mali Republic, x + code for Cote d Ivoire, and x + code for Niger. x+ x+ + x+ x + x + x x x x + + x + + The codes are: for Mali, for Cote d Ivoire, for Niger. 0. x+ x+ (+ x) x + x + x x x x () + x + () 0 The lengths of the pieces are feet, feet, and 0 feet.. Let x the number. x x x x x The number is.. Let x species of grasshoppers, then 0x species of beetles. x+ 0x 0, 000 x 0, 000 x 0, 000 x 0, 000 0x 0(0,000) 00,000 There are 00,000 species of beetles and 0,000 species of grasshoppers.. Let x the measure of the smallest angle, x + the measure of the second, x + the measure of the third, and x + the measure of the fourth. x+ x+ + x+ + x+ 0 x + 0 x + 0 x 8 x 8 x 8 x x x The angles are 8, 8,, and. 8. Let x first odd integer, then x + next odd integer, and x + third consecutive odd integer. x+ ( x+ ) + ( x+ ) x + x + x x x x + + x + + The code is,,. 0. Let x the number. ( x+ ) ( x+ ) x+ x+ x+ x+ x x x x x x 0 x The number is 0. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

26 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra. Let x votes for Zanzi, then x +,0 votes for Bishop. x + x +, 0 8, x +,0 8, x +,0,0 8,,0 x, x, x, x +,0, +,0, Zanzi received, votes and Bishop received, votes.. Let x the measure of the first angle then x the measure of the other. x+ x 0 x 0 x + 0+ x x x x () The angles are and.. Let x the number. + x x + x x + x 8x + x 8x 8x 8x + x + x x x x The number is. 8. Let x the measure of each of the two equal angles A and D, and x the measure of each of the other two equal angles C and B. x+ x+ x+ x 0 x 0 x 0 x 0 x (0) 0 The angles are A 0, D 0, C 0, B Let x floor space of Empire State Building, then x floor space of the Pentagon. x+ x 800 x 800 x 800 x x () The Empire State Building has thousand square feet and the Pentagon has thousand square feet.. Let x the number. x x 8 x The number is.. Let x first integer (smallest piece) then x + second integer (middle piece) and x + third integer (longest piece) x+ ( x+ ) + ( x+ ) 8 x + 8 x + 8 x x x x + + x The pieces measure inches, inches, and 8 inches.. Let x points for Michigan, and x + points for USC. x+ ( x+ ) 0 x + 0 x + 0 x x x 8 x Michigan scored 8 points and USC scored points. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

27 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving 8. Let x smallest angle, then x largest angles. x+ x+ x 80 x 80 x 80 x 0 x (0) 80 The angles measure 0, 80, and Let x length of first piece, then x length of second piece, and x length of third piece. x+ x+ x 0 0x 0 0x x x () x () The first piece is feet, the second piece is feet, and the third piece is feet.. The heights of the bars representing Texas and Pennsylvania are between 0 and 0. Therefore, Texas and Pennsylvania spend between $0 million and $0 million on tourism.. Let x amount spent by Pennsylvania, then x. amount spent by Hawaii. x+ (x.). x.. x x 0. x 0. x. x. (.). Pennsylvania spends $. million and Hawaii spends $ million.. Answers may vary. 8. Bh ()() 0. r t 0. Answers may vary Section. Practice Exercises. Let d 80 and r. d r t 80 t 80 t t It takes seconds or minute seconds.. Let l 0 and P 8. P l+ w w w w 80 8 w 8 w w The dog run is feet wide.. Let C 8. F C+ F F + F. The equivalent temperature is. F.. Let w width of sign, then w + length of sign. P l+ w (w+ ) + w 0w+ + w w + w + 0 w 0 w w w + () + 8 The sign has length 8 inches and width inches. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

28 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra. I Prt I Prt Pt Pt I I r or r Pt Pt. H as+ 0a H 0a as+ 0a 0a H 0a as H 0a as a a H 0a H 0a s or s a a. N F+ d( n ) N F F+ d( n ) F N F d( n ) N F d( n ) n n N F N F d or d n n 8. A a ( b+ B ) A a( b+ B) A a( b+ B) A ab+ ab A ab ab+ ab ab A ab ab A ab ab a a A ab A ab B or B a a Exercise Set.. Let d and t. d rt r() r r. Let l, w 8, and h. V lwh V (8)() V. Let A 0, B, and b. A hb ( + b ) 0 h( + ) (0) h(0) 0 0h 0 0h 0 0 h 8. Let V, and h. V Ah A() () ( A) A A A 0. Let r., and π.. Aπr A.(.) A.(0.) A.. Let I,0,000, R 0.0, and T. I PRT,0,000 P(0.0)(), 0, P, 0, P 0. 0.,00,000 P. Let r and π.. V πr (.)() V V (.)() V (8.8) V.0 (V. using a calculator.) 8 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

29 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. A πab A πab πa πa A b πa 8. T mnr T mnr mr mr T n mr 0. x+ y x + x+ y + x y + x. A P+ PRT A P P P+ PRT A P PRT A P PRT PR PR A P T PR. D fk D fk D fk D fk f f D k f. PR x + y + z + w PR ( x + y + w) x + y + z + w ( x + y + w) PR x y w x + y + z + w x y w PR x y w z 8. S lw+ wh S lw lw lw+ wh S lw wh S lw wh w w S lw h w 0. a.. a. A bh P l+ l + l P + + A P 08 A 8 The area is 8 square feet and the perimeter is 08 feet. b. The fence has to do with perimeter because it is located around the edge of the property. The grass seed has to do with area because it is located in the middle of the property. A bh A.() A. P l+ l P (.) + (.) P P The area is. square feet and the perimeter is feet. b. The border has to do with the perimeter because it surrounds the edge. The paint has to do with the area because it covers the wall.. Let A,00 and l 00. A lw, 00 00w, 00 00w w The width is feet.. Let C. F ( ) + + The equivalent temperature is F. 8. Let d 0 and t d rt 0 r 8 0 r (0) r 0 r 0 r 0 r 8. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

30 Chapter : Equations, Inequalities and Problem Solving ISM: Beginning and Intermediate Algebra 0 r The average rate was mph. 0. Let P 00 and l w 0. P l+ w 00 (w 0) + w 00 w 0 + w 00 w w w 0 w 0 w l w 0 (0) The length is 0 meters.. Let x the measure of each of the two equal sides, and x the measure of the third. x+ x+ x x x + + x x x 8 x 8 The shortest side is feet.. Let d 00 and r. d rt 00 t 00 t 00 t t 8 The trip will take hours.. Let r and h. Use π.. V πr h V (.)() () (.)()() 0. Let x number of goldfish and volume per fish. 0. x 0. x. x goldfish can be placed in the tank. 8. Let A 0, and b. A bh 0 () h (0) h 0 h 0 h 8 h The height is 8 feet. 0. Let r 000. Use π.. C πr (.)(000) C,0 The length of rope is,0 miles.. x+ (x 8) + (x ) 8 x 0 8 x x 0 x 0 x x 8 () 8 x () The lengths are feet, feet, and feet.. Let d and r 0.. d rt 0.t 0.t t 0. t It took about hours. 0 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

31 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x the length of a side of the square and x the length of a side of the triangle. P(triangle) P(square) (x ) x x x x x x x x 0 x + x x x. x (.) 0 The side of the triangle is 0 units and the side of the square is. units. 8. Let d 0 and r. d rt 0 t 0 t 0 t 0 0 t The trip will take 0 hours or hours 0 minutes. He should arrive at :0 A.M. 0. Let F 8. F C+ 8 C + (8) C + 0 C C C 0 C 0 C 0 C The equivalent temperature is C.. Let C 0. F C+ ( 0) The equivalent temperature is F. Let d 80 and r 0. d rt 80 0t 80 0t t 0 80 t.8 0 The trip will take.8 hours.. Let π. and d 0 so r. (.)() V πr,0 The volume is,0 cubic inches. 8. Let F. C ( F ) C ( ) The equivalent temperature is C. 0. Half the product of a number and five is ( x ).. Double the sum of ten and four times a number is (0 + x).. A number minus the sum of the number and six is x (x + ) Let F 0,000. C ( F ) C (0,000 ),0 The equivalent temperature is,0 C. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

32 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra 80. Let d 8,80 and r 8,000. d rt 8,80 8, 000t 8,80 8, 000t 8, 000 8, 000. t It will take. seconds. miles 8. 0 hour miles 80 feet hour 0 hour mile 00 seconds 88 feet/second 88 Let d 00 and r. d rt t 88 (00) t t It will take about. seconds. 8. Let C F. F C+ F F + F F F F + F F () F 0 The measurements are the same number at The original box has a volume V LWH. The altered box, has a length L, a width W, a height H and a new volume V L(W)(H) 8LWH. The volume is multiplied by 8. Section. Practice Exercises. Let x the unknown percent. x x 0. x The number is.% of.. Let x the unknown number. 8 % x 8 0.x 8 0.x x The number 8 is % of 0.. a. From the circle graph, % of trips made by American travelers are for combined business/pleasure. b. From the circle graph, % + % + % 8% of trips are for business, pleasure, or combined business/pleasure. c. Since % are trips for business/pleasure, find % of. 0.0 We can expect of the Americans to be traveling for business/pleasure.. Let x discount. x 8% 80 x x 08 The discount is $08. New price $80 $08 $. Increase,800 8,00 00,00 Let x percent increase. 00, 00 x 8, 00 00, 00 8, 00x 8, 00 8, x The percent increase is 0.%. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

33 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x number of new films in 00. x+ 0.08x.08x.08x x 0 There were 0 new feature films released in 00.. Let x number of liters of % solution. Eyewash No. of gallons Acid Strength Amt. of Acid % x % 0.0x % x % 0.0( x) Mix: % % 0.0() 0.0x+ 0.0( x) 0.0() 0.0x x x x x x x x She should mix liters of % eyewash with liters of % eyewash. Vocabulary and Readiness Check. No, % + % + 0% 0% 00%.. No, 0% + 0% + 0% 0% 00%.. Yes, % + % + % + % 00%.. Yes, 0% + 0% + 0% 00%. Exercise Set.. Let x the number. x 88% 000 x x is 88% of Let x the percent. 8. x 8. x 0. x 8. is 0% of. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

34 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra. Let x the number. % x 0.x 0.x x is % of Exports % Not Exports 00% % 8% 8% of corn production is not used for exports. 0. % of 0, 0. 0,.. million bushels or,,00,000 bushels were used for food, seed, or other.. Let x amount of discount. x %.0 x 0..0 x.. New price.0.. The discount was $. and the new price is $... Let x tip. x 0%.0 x 0.. x.08 Total The total cost is $8.8.. Increase,00, Let x percent. 00 x, , 000x, 000, 000 x The percent increase is %. 0. Let x original price and 0.x increase. x+ 0.x 80.x 80.x 80.. x The original price was $.. Let x last year s salary, and 0.0x increase. x+ 0.0x, 0.0x, 0.0x, x, 000 Last year s salary was $,000.. Let x the amount of % solution. No. of cu cm Strength Amt. of Antibiotic % x 0. 0.x 0% (0.) 0% x (x + 0) 0.x+ 0(0.) 0.( x+ 0) 0.x+ 0.x+ 0.x 0.x+ 0.x 0.x+ 0.0x + 0.0x + 0.0x 0.0x x Add 0 cc of % solution. 8. Decrease 00 8 Let x percent. x 00 00x x The percent decrease is %. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

35 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x the pounds of cashew nuts. No. of lb Cost/lb Value Peanuts 0 (0) Cashews x x Mix x (x + 0) (0) + x.0( x+ 0) 0 + x.x x.x.x.x x x 0 0.x 0.x 0.. x Add pounds of cashews. 8. Let x the number. x 0% 8 x. 8 x 0. 0% of 8 is Let x the number.. % x. 0.x. 0.x x. is % of.. Let x the percent. x x. x is 0% of.. From the graph, the height of the bar is. Therefore, % of the population in Charlottesville, Virginia, shop by catalog.. 8% of 0, ,8,0 We predict,0 catalog shoppers live in Juneau. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

36 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra 8. Kraft Foods North America Volume Food Produced in a Year Food Group Cheese, Meals, and Enhancers Biscuits, Snacks, and Confectionaries Beverages, Desserts, and Cereals Oscar Mayer and Pizza Volume (in pounds) Percent (Round to Nearest Percent) 8 8, % 08 Example: % 08, 0 0 8%, 0 0 %, Total, % due to rounding Source: Kraft Foods, North America 0. Let x the decrease in price. x 0.(0.) The decrease in price is $0.. The new price is $0.8.. Decrease... Let x the percent.. x...x.. 0. x The percent decrease is.%.. Let x amount produced in 00, and 0.x amount of increase. x+ 0.x 0,800.x 0,800.x 0,800.. x 00 There were 00 million gallons produced in 00.. Let x the gallons of water. No. of Gal. Strength Amt. of Antifreeze Water x 0 0 0% (0) 0% x (x + 0) 0.(0) 0.( x + 0) 0.x x x 0.x x Add gallons of water. 8. Let x mark-up. x 0%.0 x 0.. x. New price The mark-up is $. and the new price is $ Increase 8 Let x percent. 8 x 8 x x The percent increase is 00%.. Let x average in 0, then 0.x decrease. x 0.x. 0.x. 0.x x. In 0, there were. children per woman. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

37 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x ounces of self-tanning lotion. self-tanning lotion everyday lotion experimental lotion ounces $ per ounce $ x.00 x (800) x (x + 800) x+ 0.(800).( x+ 800) x+ 0.x+ 0 x+ 0.x.x+ 0.x.8x x x 0.8x x 00 Therefore, 00 ounces of self-tanning lotion should be mixed.. Increase 8 Let x percent. 8 x 8 x 0.08 x The percent increase is.8%. 8. Let x movie screens in 000, then 0.0x increase. x+ 0.0x, 0.0x, x, There were, movie screens operating in Let x increase. x 80.% 8. x x.0 His throw Christian Sandstrom s throw was 0 meters.. % of We would expect U.S. colleges to have Internet access in their classrooms.. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

38 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra. ( ) 8. ; > 0. Answers may vary. a. Yes; answers may vary b. No; answers may vary. is y percent of 00. y(00) y 0.0.% 00 This is about.% of the daily value.. Let x percent of calories from fat. x(80) () x 0..% 80.% of the calories in one serving are from fat. 8. Answers may vary Section. Practice Exercises. Let x time down, then x + time up. Rate Time Distance Up. x +.(x + ) Down x x d d.( x+ ) x.x+. x..x..x.. 0. x Total Time x + + x The entire hike took. hours.. Let x speed of eastbound train, then x 0 speed of westbound train. r t d East x..x West x 0..(x 0).x+.( x 0).x+.x x x 8 x 8 x x 0 0 The eastbound train is traveling at mph and the westbound train is traveling at mph.. Let x the number of $0 bills, then x + number of $ bills. Denomination Number Value $ bills x + (x + ) $0 bills x 0x ( x+ ) + 0x 0 x+ + 0x 0 + x 0 x x x There are 0 $ bills and $0 bills. 8 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

39 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x amount invested at.%, then 0,000 x amount invested at %. Principal Rate Time Interest.% x 0. x(0.)() % 0,000 x (0,000 x)() Total 0, x+ 0.0(0,000 x) 0 0.x x x 0 0.0x 0 0.0x x 8, 000 0,000 x 0,000 8,000,000 She invested $8,000 at.% and $,000 at %. Exercise Set.. Let x the time traveled by the bus. Rate Time Distance Bus 0 x 0x Car 0 x +. 0(x +.) d d 0x 0( x+.) 0x 0x+ 0 0x 0 0x x It will take the bus hours to overtake the car.. Let x the time to get to Disneyland and. x the time to return Rate Time Distance Going 0 x 0x Returning 0. x 0(. x) Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

40 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra d d 0x 0(. x) 0x 88 0x 0x 88 0x x. It took. hours to get to Disneyland. d rt d 0(.) 0 The distance to Disneyland is 0 miles.. The value of z quarters is 0.z. 8. The value of 0 z half-dollars is 0.0(0 z). 0. The value of z $00 bills is 00(z) or 00z.. The value of y $0 bills is 0( y).. Let x number of $0 bills, then x number of $0 bills. Number of Bills Value of Bills $0 bills x 0(x) $0 bills x 0x Total 0 0( x) + 0x 0 0x+ 0x 0 0x 0 x x () 8 There are 8 $0 bills and $0 bills.. Let x the amount invested at % for one year. Principal Rate Interest % x x 0% x (x + 0) Total 0 0.0x+ 0.0( x+ 0) 0 0.0x+ 0.0x+ 0 0.x x 0.x x 00 x She invested $0 at 0% and $00 at %. 8. Let x the amount invested at 0% for one year. Principal Rate Interest 0% x x % x 0. 0.(x) Total x+ 0.( x) x+ 0.x 80 0.x 80 0.x x 800 x (800),000 He invested $,000 at % and $800 at 0%. 0. Let x number of adult tickets, then x number of child tickets. Number Rate Cost Adult x x Child x ( x) Total, x+ ( x), x+ 0, 80 x, 0, 80 + x, x x x Sales included adult tickets and child tickets. 0 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

41 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x the amount invested at % for one year. Principal Rate Interest % x 0. 0.x % 0,000 x (0,000 x) 0.x 0.0(0, 000 x) 0 0.x x 0 0.x x x x 000 0,000 x 0, ,000 She invested $,000 at % and $000 at %.. Let x the time they are able to talk. Rate Time Distance Cade x x Kathleen x x Total 0 x+ x 0 x 0 x 0 x They can talk for hours.. Let x number of quarters, then x number of dimes. Number Value Quarters x 0.x Dimes x 0.0(x) Total. 0.x+ 0.0( x). 0.x+ 0.x. 0.x. x x () 8 The collection has quarters and 8 dimes. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

42 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra 8. Let x the amount invested at % for one year. Principal Rate Interest % x x 0% x (x) % x 0. 0.(x) Total 0 0.0x+ 0.( x) + 0.( x) 0 0.0x+ 0.x+ 0.x 0 0.x 0 0.x x 00 x (00) 000 x (00),00 She invested $00 at %, $000 at 0% and $,00 at %. 0. Let x the time it takes them to meet. Rate Time Distance Nedra x x Latonya x x Total x+ x x x x They meet in hours.. Let x the time before getting stopped. Rate Time Distance Before 0 x 0x After 0 x 0( x) Total Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

43 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving 0x+ 0( x) 0x+ 0 0x 0x + 0 0x 0x 0 0 x. He drove. hours before getting stopped.. ( ) + ( 8) ( ) 0. Let x number of quarters, then + x number of dimes, 8x number of nickels, x + number of pennies. Number Value Quarters x 0.x Dimes + x 0.0( + x) Nickels 8x 0.0(8x) Pennies x + 0.0(x + ) Total.8 0.x+ 0.0( + x) + 0.0(8 x) + 0.0( x+ ).8 0.x x+ 0.x+ 0.x x x 0. x + x + 0 8x 8() x + () + There were quarters, 0 dimes, nickels, and pennies.. R C 0x 0x x 000 0x x 00 Should sell 00 boards to break even. C R 0x 0(00) 0,000 It costs $0,000 to produce the break-even number of boards. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

44 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra. Section.8 R C 0x x 0x 0x x 0x x 80 x 80 x.8 Should sell monitors to break even. Practice Exercises. x < Place a parenthesis at since the inequality symbol is <. Shade to the left of. The solution set is (, ). ). x + x + x The solution set is [, ).. x x x The solution set is (, ].. x > x > x > The solution set is (, ). (. x x x x x The solution set is [, ).. x+ 0 x+ x+ 0 x x+ x x + 0 x x The solution set is (, ].. x > ( x ) x > x x x > x x 8x > 8x > 8x > 8 8x 8 < 8 8 x < The solution set is,. ) 8. ( x ) ( x ) x x x x x x x x x x + + x 0 x 0 x 0 The solution set is [0, ). 0. x < Graph all numbers greater than or equal to and less than. Place a bracket at and a parenthesis at. The solution set is [, ). ) Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

45 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving 0. < x + 8 < x + 8 < x x < < x The solution set is (, ]. ( ]. < x + < () < x + () < < x + 0< 0 < x < 0 < x < x < < < x < The solution set is,. ( 0 (. Let x number of classes x x x 00 x 00 x. Kasonga can afford at most community college classes this semester. Vocabulary and Readiness Check. x (x + ) is an expression.. x (x + ) is an equation.. x < (x + ) is an inequality.. y 8 is an inequality... x + is an equation. x + is an expression.. is not a solution to x. 8. is not a solution to x <... is not a solution to x < is not a solution to x. Exercise Set.8. (, ), x > (. (, ], x. y < 0, (, 0) ) 8. z <,, ) 0 0. x >, (, ) (. x > x >, (, ) (. x + x, (, ]. x < 0 x 0 > x >, (, ) ( 8. x 0 8x + x 0 x, [, ) Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

46 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra 0. x+ < x x x+ < x x + < 0 x <, (, ) ). x+ ( x ) x+ x x + x x x, [, ). x+ > ( x) x+ > x x + > x > 8 x 8 < x <, (, ) ). (x ) (x ) x x 8 x 8 x x,, 8. ( x ) + x ( x) x + x 0+ x 8x + x x x 8 x,, 0. x > x < x <, (, ) ). y y, (, ]. x x x x x x, (, ]. x < 8x+ x x < 0x x < 0 x < x > x >,, ( 8. x x,, 8 0. ( x+ ) < (x+ ) x+ 0< 8x+ x + 0< x < 8 x 8 > 8 8 x >,, ( 8 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

47 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. ( x) x x 0 x 0 x 0, (, 0]. x+ < ( x+ ) x+ < x x + < x < x > x >,, 0 (. ( x) + x ( x) + + x+ x + x+ + x x + x 0 x x,, 8. ( x ) < x + < x < 0 x 0 > x > 0, (0, ) ( 0 0. Answers may vary. y, [, ] [. x, [, ] [ [ [. < x < < x <,, ( ( 8. x 0 x x,[, ] [ 0. 0 < ( x + ) 8 0 < x < x <, (, ] [ x ( ]. < + x < x < x,, ( ]. ( x + ) < 8 x + 8< 8 x < 0 x < 0,, 0 ). Answers may vary 8. x + 0 x x 0 0. Let x the number of people invited. x x 80 x. They may invite people. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

48 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra. x+ x+ 8 x + 8 x x x can be no greater than.. Let x the sales per month and x 000 sales over ( x 000) x x x 0 x,000 He must sell $,000 or more per month.. Let x the height of the center in inches x x () + x 8 x 8 He must be at least 8 in. '" tall. 8. EF-0 y 8 EF- 8 y 0 EF- y EF- y EF- y 00 EF- y Let x the unknown number. < x < < x < < x < All numbers between and 8. Let x score needed on the final exam x x (0) (00) 0 + x x 0. He must make between 8. and () ()()() 8. 0 (0)(0)(0)(0)(0) Read the value on the vertical axis corresponding to 00; $0.0. Find the points on the graph greater than $. The years are 00, 00, 00.. C.d 8.d.8 d 8.8 The diameter must be between.8 mm and 8.8 mm.. xx ( ) x x 8 x x x x 8 x x 8 x 8 x, [, ) 8. x x+ 8 < x( x+ 8) x x+ 8< x + 8x x+ 8< 8x x + 8 < 0 x < 8 x 8 > x >,, ( The Bigger Picture. x x x The solution is.. x > x < x < The solution set is (, ). 8 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

49 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. y y + + y y y The solution is.. x x x x x x x x + + x x x The solution is.. ( x ) x+ x 8 x+ x 8 x x+ x x 8 x x x x The solution set is [, ).. (x ) (0 x ) 0x 0x 0x 0x 0x 0x Since this is a false statement, there is no solution... 0.x x x. 0.x x The solution is. 8. ( x ) < ( x+ ) ( x ) ( ) < x+ ( x ) < ( x+ ) x < x+ x x < x+ x x < x + < + x < The solution set is (, ).. y ( y ) ( y 0) y y+ 0 y+ 0 y+ 0 y+ 0 y+ 0+ y y+ 0+ y 0 0 All real numbers are solutions. 0. ( x ) ( x+ ) ( x ) ( x+ ) ( x ) ( x+ ) x x+ x x x+ x x x + + x x x The solution is. Chapter Vocabulary Check. Terms with the same variables raised to exactly the same powers are called like terms.. A linear equation in one variable can be written in the form ax + b c.. Equations that have the same solution are called equivalent equations.. Inequalities containing two inequality symbols are called compound inequalities.. An equation that describes a known relationship among quantities is called a formula. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

50 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra. A linear inequality in one variable can be written in the form ax + b < c, (or >,, ).. The numerical coefficient of a term is its numerical factor. Chapter Review. x x + x x. 0.z.x.z.x.z.. 8 x+ + x x x 0 y+ + y+ y+ y+. (n ) + n 0 n 8 + n 0 n 8. ( w+ ) ( w) w+ + w w. (x + ) (x ) x + x + x + 8. ( y 0.) (.y ) y 0..y+ 0.y +.. Three times a number decreased by is x. 0. Twice the sum of a number and.8 added to times a number is (x +.8) + x.. 8x+ x 8x+ 8x x 8x x. y y y y y y y. x+ x x x. x x+ x + x. x x x x 0. ( x+ ) ( + x) x+ + x x + x. ( + n) ( n ) 8 + n n 8 + n n 8. ( + x) (x+ ) ( x ) x x x+ 0 + x+ x+ x + x 8. x x + + x 8 0. x + x + x. 0 x; choice b.. x ; choice a.. Complementary angles sum to 0. (0 x) ; choice b.. Supplementary angles sum to (x + ) 80 x x ( x) ; choice c... x x ( ) x x x x. x 0 x 0 x 0 80 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

51 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving 8. y y y. 0.x 0. 0.x x 0. x 0. x x. x + x 8 x 8 x. x + 0 x x x. ( x ) + x x + x x+ x x x x. x x x x x x. 0 x + 0 x + x + 0 x x x. x+ x + x + x x+ x x. Let x the first integer, then x + the second integer, and x + the third integer. sum x + (x + ) + (x + ) x + 8. Let x the first integer, then x + the second integer x + the third integer x + the fourth integer. sum x + (x + ) x x+ x x+ x x+ x x x x+ x x+ 8 x 8 8 x+ 8 x 8 x x. (x+ ) x+ x x+ x x x Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall 8

52 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra. (x+ ) x+ 8x x+ x x x. (x ) ( + x) x+ 0 x 0 There is no solution.. (8y ) ( + y) y 0 + y 0 There is no solution... ( z) z ( z) z z z 8z z 8 z ( n + ) n ( n+ ) n n+ 8 n 8 n 8 n. 0.(n ) 0. 0.( + n) 0[0.(n ) 0.] 0[0.( + n)] (n ) ( + n) 0n + 8n 0n + 8n n n 0 n 0 8. a (a ) a 8a a a a. 0. ( c + ) c ( c+ ) (c ) c+ c 8 c + 8 c c (8 a) a (8 a) ( a) a a 0a + 0a a 0 a. 00(0x 0) (0x,00), 000x, 000,80x+,, 00 0,80x, 000,, 00 0,80x,, 00 x 0..y 0.0y 0..8y 0. y 0.. Let x length of a side of the square, then x the height. x+ ( x) x + 0. x. x x (.). The height is. inches.. Let x the length of the shorter piece and x the length of the other. x+ x x x x () 8 The lengths are feet and 8 feet. 8 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall

53 ISM: Beginning and Intermediate Algebra Chapter : Equations, Inequalities, and Problem Solving. Let x number of Keebler plants, then x number of Kellogg plants. x+ (x ) x x x 8 x (8) There were 8 Keebler plants and Kellogg plants.. Let x first integer, then x + second integer, and x + third integer. x+ ( x+ ) + ( x+ ) x + x x x x + + The integers are, 8,.. Let x the unknown number. x x x ( x ) x x x x The number is. 8. Let x the unknown number. ( x+ ) x x+ x x x The number is.. Let P and l. P l+ w () + w 8 + w 8 w w 0. Let V, l 8, and w. V lwh 8() h 8h h. y mx+ b y b mx y b m x. r vst r+ vst r + s vt. y x x y+ y + x y x. x y y x x y x + y. C πd C π D. C πr C π r. Let V 00, l 0, and h. V lwh 00 0 w() 00 0w w The width is meters. 8. Let x width, then x + length. 0 x+ ( x+ ) 0 x+ x+ 0 x + 8 x x x The dimensions are 8 feet by feet. Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall 8

54 Chapter : Equations, Inequalities, and Problem Solving ISM: Beginning and Intermediate Algebra. Let d 0,000 and r. d rt 0, 000 t 80 t It will take 80 minutes or hour and 0 minutes. 0. Let F 0. C ( F ) (0 ) () 0 The temperature was 0 C.. Let x the percent. x x 0. x is 0% of.. Let x the percent.. x 8. 8x x. is 0% of 8.. Let x the number.. % x..x..x.. 0 x. is % of 0.. Let x the number. 8 0% x 8 0.x 8 0.x x 8 is 0% of 80.. Let x mark-up. x % 00 x x 0 New price The mark-up is $0 and the new price is $0. 8 Copyright 00 Pearson Education, Inc., publishing as Pearson Prentice Hall