Chapter 1: Review of Real Numbers

Transcription

1 Chapter : Review of Real Numbers Prep Test a, c, d 0a. 0. C 0b D 0c A 0d B Go Figure Since the other three smaller rectangles are of different sizes, the possible values for x cannot be the same as any of the three other rectangles. The first possible value of x is. The lengths of the sides of the rectangles are shown below. The second possible value of x is 4. The first possible value of x is 4. The lengths of the sides of the rectangles are shown below. The third possible value of x is 9. The lengths of the sides of the rectangles are shown below. Section. Concept Review.. Sometimes true The only exception to this statement is for zero. 0 0, which is not a positive number.. Sometimes true If x is a positive integer, then x is a negative integer. If x is a negative integer, then x is a positive integer.. Never true 4 < but ( 4) is not less than ( ). 7. Always true Objective.. Exercises. 4: c, e 9: a, b, c, d 0: b, c : a, b, c, d 7.8: none 66: c, e. : b, d 0: a, b, d : a, b, d π: c, d.: b, d 4.: c, d 4 : c, d 7: c, d Replace x with each element in the set and determine whether the inequality is true. x < < True 0 < True 7 < False The inequality is true for and 0.

2 Chapter : Review of Real Numbers. Replace y with each element in the set and determine whether the inequality is true. y > 4 6 > 4 False 4 > 4 False 7 > 4 True The inequality is true for 7.. Replace w with each element in the set and determine whether the inequality is true. w True True 0 False False The inequality is true for and.. Replace b with each element in the set and evaluate the expression. b ( 9) 9 (0) 0 (9) 9 7. 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.. Exercises. {,, 0,,,, 4}. {, 4, 6, 8, 0, } 7. {, 6, 9,,, 8,, 4, 7, 0} 9. {, 0,, 0,, 0, } 4. { x x > 4, x is an integer} 4. { x x } 4. { x 0 < x <} 47. { x x 4} 49. A B {,, 4, 6, 9}. A B { 4,, 0,, 4, 8}. A B {,,, 4, } 7. A B {6} 9. A B {, 0, 0} 6. A B 6. A B {4, 6} 6. { x < x < } 67. { x 0 x } 69. { x x <} 7. { x x } 7. { x x >} { x x < } 7. { x x } { x x 0} 77. { x x >} { x x } 79. { x x >} { x x >} 8. { x 0 < x < 8} 8. { x x 7} 8. { x x < 6} 87. { x x 4} 89. { x x >} 9. (, 4) 9. [, ] 9. (, ) 97. [, 6) 99. (, ) 0. (, ). Α B {,,, 8, 9, 0}

3 Section. 0. [, ] 0. (, ] 07. [, ) 09. (, ] [4, ). [, ] [0, 4]. (, ) (, 4] Applying Concepts.. A B is { x x } { x 0 x } { x x } A 7. B B is set B. 9. A R is { x x }, which is set A.. B R is the set of real numbers, R.. R R is the set R.. B C is { x 0 x } { x x 0}, which contains only the number A B { x x > 0, x is an integer} 7. A B { x x, x is an odd integer} 9. The answer is b and c. For example: 4 a. 0 0 False b True 4 c. 0 0 True d False Section. Concept Review.. Sometimes true ( ) + 4, a positive number ( 8) + 4 4, a negative number. Never true Always true 7. Never true The Order of Operations says to work inside parentheses before doing exponents. 9. Always true Objective.. Exercises. 8 + ( ) ( ) ( 8) ( 8) ( ) ( ). 0()( 6) 700( 6),00 7. ( 7)( 6) 98,9 9. ( 8) ( 8) ( 9) 7 7

4 4 Chapter : Review of Real Numbers ( 6) 4 0 +( 6)+ ( 4)+ ( ) 46 +( 4)+( ) 60 +( ) 6. +( 9) 6+ +( 9) +( 6)+ + ( 6) ( ) ( 6) ( 0) 78 +( 46) +( 0) 69 +( 0) ( 7) Objective.. Exercises / / / 7 / / / / / / / / / / / / / / / / / 7 / / / / (0.0)(0.)(6.) (0.)(6.) ) ( 0.9) ) [ ( 6.07)] ( 7.4) 6.4 +( 7.4)

5 Section. Objective.. Exercises ( ) ( ) ( )( )( ) 8. 4 ()() ()()()() ()() ()() ( ) ( ) ( )( )( ) ( )( ) ()()() ()()() ( ) 6 ( 8) 6 ( 8) 6 ( 6) [ ( 4 +) ] 6[ ( ) ] 6[ ( )] 6[+] 6[4] ( 0)( ) ()() ( 0) ( )( ) 4 ( 0)(4) 40(4) ( 7) ( ) ( ) 4 (8) ,76 Objective..4 Exercises 97. (8 4) ( ) (4) [( 4) ] [( ) ] [ 6 ] [ 8] (..) (.) (.) (.)

6 6 Chapter : Review of Real Numbers (6.4.7).76 () Applying Concepts.. 0. No, the multiplicative inverse of zero is undefined.. 7 8,68,4,97,90,449 The ones digit is has over 0 digits. The last three are Find b c. Then find a (bc ). Section. Concept Review.. Sometimes true The reciprocal of is, a whole number. The reciprocal of is, not a whole number.. Sometimes true xy and xy are like terms with the same variables. xy and x y are unlike terms with same variables.. Always true Objective.. Exercises ( + 4) + + (4 + ). is undefined (x + ) x (mn) mn. (x) ( ) x. A Division Property of Zero 7. The Inverse Property of Multiplication 9. The Addition Property of Zero. A Division Property of Zero. The Distributive Property. The Associative Property of Multiplication Objective.. Exercises 9. ab + dc ()() + ( 4)( ) cd a 4( )( 4) () 4( )( 4) 4 ( 4)( 4) (b a) + c [ ()] +( ) [ 4] + ( ) [ ] + ( ) +( ) 0. (bc + a) (d b) [()( ) +] ( 4 ) [ + ] ( 7) [ ] ( 7) a4 6 bc 4 ()4 6 ()( ) 4 (6) 6 ()( ) ac 4 c ()( ) ()( ) 4 ( ) ( ) 6( ) 4 ( ) 6 4 ( ) ( ) b c a c () ( ) () ( ) 9 ( ) 6 ( ) a d b + c ( 4) +( ) +4 + ( ) 6

7 Section aa+ d + ( 4) +( 8) 6 (6) 47. a 4d b c () 4( 4) 4 ( 6) () ( ) 9 ( ) d ( 4) ab 4c b + c () 4( ) ( ) + ( ) ( 4) 6 ( 4) 6 + ( ) ( 4) ( ) ( 4) 0 ( 4) ( 4) 6. (d b) (a c) ( 4 ) [() ( )] ( 7) [6 ( )] ( 7) [6 +] ( 7) d c a ( 4) ( ) () 6 ( )() d + 4ac ( 4) +4()( ) ( 64) +8( ) (a) 4 ( ) V LWH V (4)(0)(6) V 840 The volume is 840 in. 6. V s h V ( ) V The volume is ft. 6. V 4 πr r V 4 d (). π(.) V 4.π V 4.4 The volume is 4.π cm. The volume is approximately 4.4 cm. 6. SA LW + LH + WH SA ()(4) + ()() l+ (4)() SA SA 94 The surface area is 94 m. 67. SA s + 4 bh SA 4 + ( 4)( ) SA The surface area is 6 m. 69. SA πr + πrh SA π(6 )+ π(6)() SA 7π+4π SA 96π SA 0.9 The surface area is 96π in. The surface area is approximately 0.9 in. Objective.. Exercises 7. x + 0x x 7. x + x 7x x 7x 4x 77. a + 7b + 9a 7a + 7b 79. x x 8. (x ) x (x + ) x ( x y) x + y 87. (x y) x 6y 89. a (a 7) a 9a + a + 9. x (x y) x x + 6y x + 6y 9. [ 6( a )] [ 6a +0] [8 6a] 40 0a 9. [ y ( y x)] [y y + 6x] [ y +6 x] 0y+ 0x 97. 4( a b) (a b) 4a 8b 6a +0b 0a + b 99. 7(a b) +( b + a) 4a + 7b 6b + a a + b

8 8 Chapter : Review of Real Numbers 0. x 4[ x 4( y [y +])] x 4[ x 4( y 0y 6)] x 4[ x 4( 9y 6)] x 4[ x + 6y +4] x 4x 44y 96 x 44y x + 8(x 4) (x y) x + 8x 6x + y x + y 0. 4 [4x ( x 8) 7x] [4x x +4 7x] 4 [4x + 4] 4 x +6 Applying Concepts (y + ) y + 4 The statement is correct; it uses the Distributive Property x ( + )x x The statement is not correct; it mistakenly uses the Distributive Property. It is in an irreducible statement. That is, the answer is + x.. (y) ( )(y) y The statement is not correct; it incorrectly uses the Associative Property of Multiplication. The correct answer is ( )y 6y.. x + y y x The statement is correct; it uses the Commutative Property of Addition.. a + 4(b + a) a. a + (4b + 4a) Distributive Property b. a + (4a + 4b) Commutative Property of Addition c. (a + 4a) + 4b Associative Property of Addition d. ( + 4)a + 4b Distributive Property 7a + 4b 7. (a + ) Section.4 a. (a) + () Distributive Property b. ( )a + () Associative Property of a + () Multiplication c. a + Multiplication Property of One Concept Review.4. Never true The smaller number is represented by x.. Never true The sum of twice x and 4 is represented by x Sometimes true The square of x is represented by ( x). The only exception is for the number 0. 0 ( 0) 0 Objective.4. Exercises. the unknown number: n The sum of the number and two: n + n (n + ) n n. the unknown number: n one-third of the number: n four-fifths of the number: 4 n n + 4 n n + n 7 n. the unknown number: n the product of eight and the number: 8n (8n) 40n 7. the unknown number: n the product of seventeen and the number: 7n twice the number: n 7n n n 9. 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 + 0n +. 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 + ) x + (7 x) x + 7. Let the larger number be x. Then the smaller number is 4 x. The difference between two more than the smaller number and twice the larger number [(4 x) + ] x 4 x + x 6 x Objective.4. Exercises 7. The population of Milan, Italy: P The population of San Paolo, Brazil: 4P 9. Distance from Earth to the moon: d Distance from Earth to sun: 90d

9 Chapter Review Exercises 9. Amount of the first account: x Amount of the second account: 0,000 x. Flying time between San Diego and New York: t Flying time between New York and San Diego: t. 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 that of angle A: (x) 4x Applying Concepts.4 7. The sum of twice a number and three. 9. Twice the sum of a number and three. One-half the acceleration due to gravity: g Time squared: t The product: gt. The product of A and v : Av Chapter Review Exercises. 4 : Replace x with the elements in the set and determine whether the inequality is true. x > 4 > False > False 0 > True > True The inequality is true for 0 and.. p { 4, 0, 7} p {,, 0,,, }. { xx< } 6. { x x } 9. [, ) 0. { x x <}. { x x } { x x > 0}. (, 4]. 0 ( ) ( 8) 7 + ( 8) ( 7) (4 ) ( ) / 8 / 8 / / 7 / / / 7 / (.06). 7. A B {,,, 4,, 6, 7, 8} 8. A B {, }

10 0 Chapter : Review of Real Numbers. a b a ( ) () ( ) 6 ( ) ( ) ( 9) ( ) (a b ) ab (4 ( ) ) (4)( ) (4 (9)) (4)( ) (4 8) (4)( ) 4 [(4)( )] y 7. (ab) The Inverse Property of Addition 0. The Associative Property of Multiplication. (x ) + 4( x) x x 6x y [ x ( x) 4y] 4y [ x 6 +4x 4y] 4y [ x 6 4y] 4y x +8 +y 6y x +8. The unknown number: x The sum of the number and four: x + 4 4(x + 4) 4x The unknown number: x The difference between the number and two: x Twice the difference between the number and two: (x ) (x ) + 8 x x + 4. Let x be the smaller of the numbers. Then the larger number is 40 x The sum of twice x and five more than 40 x. x + (40 x + ) x Let x be the larger number. Then the smaller number is 9 x. The difference between three more than twice (9 x) and one more than x. [(9 x) +] (x +) 8 x + x x The width of the rectangle: W The length is feet less than W. The length is W. 8. Let the first integer be x. The second integer is five more than four times x. 4x + is the magnitude of the second integer. Chapter Test.. Replace x with each element in the set and determine whether the inequality is true. > x > True > False > 7 False The inequality is true for.. ( ) ( ) ( ) 7 +( ) 4. ( )( )( ) (6)( ) ( ) ( ) ( ) ( 8)(9) / / / 7 / / / (8) ( ) 8 4( ) 8 4()

11 Chapter Test. (a b) (b +) ( ( )) (( ) +) () ( 6 +) () ( ) ( ) 6. b c a c () ( ) ( ) 9 ( ) The Distributive Property 9. x (x y) (y 4x) x x + y y + x x y 0. x 4[ (x + 4y) ] x 4[ x y ] x 4[ x y] x + x + 48y 4x + 48y. 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 n. The unknown number: n The total of twelve times the number and twentyseven: n + 7 (n + 7) 4n +9. A B {,,, 4,, 7} 4. A B {,, 0,,, }. A B {, 7} 6. A B {, 0, } 7. (, ] 8. (, ) 9. { x x } { x x < } 0. { x x < } { x x > }