Chapter : Review of Real Numbers PREP TEST. a..; C....... 7 + + + 9 8 7 8. 9.. + 7. 9. 8.7 7. 9 b. 7.7; D. c..7; A. d. 89.89; B. GO FIGURE If the areas of the known rectangles are,, and, the corresponding rectangles have to be sizes x, x, and x (a rectangle of size x would not meet the perpendicular line requirement). In order for the th rectangle to be attached to the three above sized rectangles, it must be of size x, x, or x. Therefore, the possible values of x, or the areas of these three rectangles are,, and 9. Answer:,, 9 SECTION. Objective A Exercises. Integers:, Rational numbers:,,,. Irrational numbers: p,., Real numbers: all. 7., 7 7..9(.)7. 7. 8... 9.. 9. Replace x with each element in the set and determine whether the inequality is true. x < < True < True 7< False 9. a. - > -8; b. - < -8; c. > -8; d. 8 > -8; a, c, d
Chapter : Review of Real Numbers. Replace y with each element in the set and determine whether the inequality is true. y > > False > False 7> True 7. Replace w with each element in the set and determine whether the inequality is true. w True True False False 9. Replace b with each element in the set and evaluate the expression. b ( 9) 9 () (9) 9. Replace c with each element in the set and evaluate the expression. c. Replace m with each element in the set and evaluate the expression. m Objective B Exercises. {,,,,,, } 7. {,,, 8,, } 9. {,, 9,,, 8,,, 7, }. {,,,,,, }. { xx>, x integers}. { xx, x real numbers} 7. { x < x<, x real numbers} 9. { x x, x real numbers} Objective C Exercises. A B {,,,,9 }. A B {,,, 8, 9,}. A B {,,,,, 8} 7. A B {,,,, } 9. A B { }. A B {,, }. A B. A B {, } 7. { xx < } 9. { xx }. { x < x < }. { x x }. { xx> } { xx< } 7. { xx } { xx } 9. { xx> } { xx } 7. { xx> } { xx> } 7. { x < x < 8}
Section. 7. { x x 7} 77. { x x < } 79. { xx } 8. { xx > } 8. (, ) 8. [, ] 87. ( 8, ) 89. [, 8) 9. (, ) 9. [, ] 9. ( 8, ] 97. [, 8) Applying the Concepts A B { x x } { x x } 99. { x x } A. B B is set B.. A R is { x x }, which is set A.. B R is the set of real numbers, R. 7. R R is the set of real numbers, R. 9. B C is { x x } { x x }, which contains only the number. SECTION. Objective A Exercises. a. Students should paraphrase the rule: Add the absolute values of the numbers; then attach the sign of the addends. b. Students should paraphrase the rule: Find the absolute value of each number; subtract the smaller of the two numbers from the larger; then attach the sign of the number with the larger absolute value.. 8 + ( ). + ( ) 7 7. ( 8) ( 8) 9 9. 8 ( ). ( ). ()( ) 7( ),. ( 7)( ) 98,9 7. ( 8) 9 9 9. ( 8) + 8. 8 7 7. ( 9) 7 7. 8 + 8+ 7. - + (-) - + (-) + (-) + (-) - + (-) + (-) - + (-) - 9. + ( 9) + + ( 9)+ ( )+ + ( )+ 7 +. + ( ) + ( ) 7. 78 +( ) 9 78+ ( )+ ( ) +( 9) 9 +( ) +( 9) 87+ ( 9) 8. ( 7) 7. 897 9 8
Chapter : Review of Real Numbers Objective B Exercises 9. a. The least common multiple of two numbers is the smallest number that is a multiple of each of those numbers.... 7. 9.... b. The greatest common factor of two numbers is the largest integer that divides evenly into both numbers. 7 + 8 8 + 8+ 8 8 8 9 7 + 9 7 + + + + + 8 7 + + + / / / 7 / 8 8 8 / / / / / / 7 7 7 / / / / / / 7 7. 7 8 7 8 / / / / 7 / / / 7 / 9. -.7 +.9 -.97. 7.8 +.8.8. (.)(.)(.) (.)(.).9. 7...9).8 8 8.8 (.9)..7.). 9...7 9. 8. [ (.7)] 7. 8..7+ ( 7.). 7. 87.9.. Objective C Exercises 7. 7. ( ) 8 77. ( ) ( )( )( ) 79. ()() ()()()() 8 8. ()() ()() 9 8. ( ) ( ) ( )( )( ) ( )( ) 8 9 7
Section. 8. ()()() ()()() 8 7 7 8 87. ( )( ) ( )( )( ) ( )() () 89. ( ) 9. ( ) (8) 9,,8 Objective D Exercises 9. We need an Order of Operations Agreement to ensure that there is only one way in which an expression can be correctly simplified. 9. (8 ) () () 7 97. 99. + + 9+ + 77. [( ) ] [( ) ] [ ] [ 8]. 8 ( ) ( 8) ( 8) ( ) +. [ ( + ) ] [ ( ) ] [ ( )] [+ ] [] 7. 9 + 9 + + + + 9. 7 7 7 7 7. 8 + 9 + 9 9 7 8 7 7 97 7..(..) +.8.(.) +.8.(.) +.8.8+.8.8..7. (.).7.. 7..7 7..7 (.9.7).7 (.9).7..89878 Applying the Concepts 9.. No, the number zero has a multiplicative inverse that is undefined.. 7 8,8,,97,9,9 The ones digit is 9.. has over digits. The last three are. 7. First find b c, then find a bc.
Chapter : Review of Real Numbers SECTION. Objective A Exercises.. ( + ) + + ( + ). is undefined. 7. (x + ) x + 9.. mn (mn). (x) ( ) x. The Division Property of Zero 7. The Inverse Property of Multiplication 9. The Addition Property of Zero. The Division Property of Zero. The Distributive Property. The Associative Property of Multiplication Objective B Exercises 7. ab + dc ()()+ ( )( ) + 9. cd a ( )( ) () ( )( ) ( )( ). (b a) + c [ ()] + ( ) [ ] + ( ) [ ] + ( ) + ( ). (bc+ a) (d b) [()( )+ ] ( ) [ + ] ( 7) [ ] ( 7) ( 7) 7. a 7. ac c 9. b c a c bc () ()( ) () ()( ) ()( ) ( ) + 9 ()( ) ( ) ( ) ( ) ( ) ( ) () ( ) () ( ) 9 ( ) ( ) 9+ + 7. a d ( ) b+ c + ( ) + + ( ). aa+ d + ( ). a d b c + ( 8) () () ( ) () ( ) ( ) 9 ( ) + 9+
Section. 7 7. ab c () ( ) d ( ) b+ c () + ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) + + ( ) 9. (d b) (a c) ( ) [() ( )] ( 7) [ ( )] ( 7) [+ ] ( 7) 7 7. d c a ( ) ( ) ( ) +. d + ac ( ) + ()( ) ( ) +8( ) 8. (a) () Objective C Exercises 7. x + 7x x 9. 8ab ab ab. x x + 9x x + 9x 7x. b 8a b 8a 7b. (y) y 7. (x 9) x + 9. (x + y) x y 7. (a ) a 7. x (y ) x y + 7. x (x 7) x x + 7x + 77. 79. [ a ( )] a [ a + a] [a ] 8a 7 [ x ( x+ y)] [ x x y] [ x y] x y 8. (x y) +(y x) x +y +y x x +y 8. (a b) ( a+ b) a b+ 8a b a b 8. x [y (x +[x + y])] x [y (x + x +9y)] x [y (7x +9y)] x [y x 8y] x [ 7y x] x +y +8x x + y 87. (7x y) ( x + y) x + y +x 9y 8x y 89. [8x (x )+ ] [8x x + + ] Applying the Concepts [x +7] x + 9 9. (y + ) y + The statement is correct; it uses the Distributive Property. 9. + x + ( + )x The statement is not correct; it mistakenly uses the Distributive Property. It is an irreducible state. That is, the answer is + x. 9. (y) ( )(y) y The statement is not correct; it mistakenly uses the Associative Property of Multiplication. The correct answer is ( )y y. 97. x + y y x The statement is correct. it uses the Commutative Property of Addition. SECTION. Objective A Exercises. The unknown number: n Eight less than a number: n 8. The unknown number: n Four-fifths of a number:. The unknown number: n The quotient of a number and fourteen: n 7. The unknown number: n The sum of the number and two: n + n (n+ ) n n n
8 Chapter : Review of Real Numbers 9. The unknown number: n The product of eight and the number: 8n (8n) n. The unknown number: n The product of seventeen and the number: 7n twice the number: n 7n n n. The unknown number: n The square of the number: n The total of twelve and the square of the number: + n n ( + n ) n n. The unknown number: n The sum of five times the number and : n + The product of the number and fifteen: n n+ (n+ ) n+ n + n + 7. Let the smaller number be x. The larger number is x. The sum of twice the smaller number and two more than the larger number. ( x+ ) + x (7 x) + x x + 7 9. Let the larger number be x. The smaller number is x. The quotient of five times the smaller number and the difference between the larger number and three. ( x) x Objective B Application Problems. The distance from Earth to the moon: d The distance from Earth to the sun is 9 times d: 9d. The amount of caramel in the mixture: c The mixture contains lb more of chocolate than of caramel: c +. The amount in the first account: x The total amount is,. The amount in the second account:, x 7. The measure of angle B: x The measure of angle A is twice that of angle B: x The measure of angle C is twice the measure of angle A: (x) x Applying the Concepts 9. The sum of twice x and.. Twice the sum of x and.. a. One-half the acceleration due to gravity: g Time squared: t ; The product: gt b. The product of m and a: ma c. The product of A and v : Av CHAPTER REVIEW. {,,,,, }. A B {,}. (, ]. The Associative Property of Multiplication..7+..7.77.7.8. (a b ) ab ( ( ) ) ()( ) ( (9)) ()( ) ( 8) ()( ) [()( )] 7 7. ( ) ( ) 9 9 88 8. y [ x ( x) y] y [ x + x y] 9. y [x y] y x+ 8+ y y x + 8 ; +. { xx<, x real numbers}. { xx < }. ( ) 8 + + ( 8) 7+ ( 8). + + 8 +8 7 7.
Chapter Test 9. 8 8 / 8 / 8. Replace x with the elements in the set and determine whether the inequality is true. x > > > 7. a 8. False False > True > True b a ( ) ( ) () ( ) ( ) (9) ( ) 8+ 8 + 8 8 8 9. 9 + 9+. [, 8). A B {,,,,,,7,8}. ( 7). { x x }.. 7 7 / / / 7 / / / 7. { xx } { xx> } 7. (x )+ ( x) x + + 8 x x + 8. p {,,7} p 7 7 9. The Inverse Property of Addition..8 (.).. The unknown number: x Three more than a number: x + The quotient of three more than a number and the number: x + x x + x x x x + x x x x x x. Let x be the smaller of the numbers. Then the larger number is x. x + ( x)+ x +. The width of the rectangle: w The length is feet less than w. The length is w. The number of times Reagan used the presidential veto: x The number of times Carter used the presidential veto: x 7 CHAPTER TEST. ( )( )( ) ()( ). A B {,7}. ( ) ( ) ( 8)(9) 7. ( 8, ]. A B {,, }. (a b) (b+ ) ( ( )) (( ) + ) () ( + ) () ( ) ( ) 7. ( ) +
Chapter : Review of Real Numbers 8. x [ (x + y) ] x [ x y ] x [ x y] x + x + 8y x + 8y 9... { xx< } { xx> }. (, 8). A B {,,,,, 7}. x ( x y) ( y x) x x+ y y+ x x y. ( )+ + + + ( ) + + ( ) 7+ ( ).. + 9 + +. 9 7 / / / / / / 7. Replace x with each element in the set and determine whether the inequality is true. x < < True < False 7< False 7. b c () ( ) a c ( ) 9 ( ) 8 8. 8. 8 ( ) 8 ( ) 8 () 8 8. 7 / / / 7 / 7 / /. The Distributive Property. { xx } { xx< } 7..7.98 +..7 +.. 8. A B {,,,,,} 9. The unknown number: n Three less than the number: n The product of three less than the number and nine: (n )(9) (n )(9) 9n + 7 9n. The larger number: x The smaller number: 9 x The difference between one more than the larger number and twice the smaller number. (x +) (9 x) x + 8+ x x 7 9. (8)