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1 7. Since the numbers have like signs, the quotient is positive = 1 9. Since the numbers have unlike signs, the quotient is negative. 4/ (-4) = 4-4 = Since the numbers have unlike signs, the quotient is negative = Since the numbers have unlike signs, the quotient is negative. 6-4 = Since the numbers have like signs, the quotient is positive (-8) = = Zero divided by any nonzero number is zero. 4 = 69. Since the numbers have unlike signs, the quotient is negative. -1 = Since the numbers have unlike signs, the -1 quotient is negative. = = Ë Ë - = 1 1 Ë - = 1 1(-) = = -1 Ë = = (-1)(1) (1)(-1) = -1-1 = = = (-1)( 1) ( 1) ( ) = -144 = Since the numbers have unlike signs, the product is negative. 4(8) = 8. Since the numbers have unlike signs, the quotient is negative. 1 - = - 8. Since the numbers have unlike signs, the product is negative. -7 ( ) = Since the numbers have unlike signs, the quotient is negative. 7 (-) = 7 - = Since the numbers have unlike signs, the quotient is negative. -1 = Since the numbers have unlike signs, the quotient is negative. 6-6 = = ( - ) = = ( ) ( ) = - -6 = 6 9. Zero divided by any nonzero number is zero. 6 = 6 = 9. Any nonzero number divided by zero is undefined. is undefined. 97. Zero divided by any nonzero number is zero. 1 = 18

2 99. Any nonzero number divided by zero is undefined. 8 is undefined. 11. a. Since the numbers have unlike signs, the product will be negative. b. 9(-8) = -496 c. Yes; as expected the product is negative. 1. a. Since the numbers have unlike signs, the quotient will be negative. b. -4 / 1 = -4 1 = -16 c. Yes; as expected the quotient is negative. 1. a. Since the numbers have unlike signs, the quotient will be negative. b. 4 (-7) = 4-7 = -9 c. Yes; as expected the quotient is negative. 17. a. Since the numbers have like signs, the product will be positive. b. (-49)(-16) = 6174 c. Yes; as expected the product is positive. 19. a. The quotient will be zero; zero divided by any nonzero number is zero. b. = c. Yes; as expected the answer is zero a. Undefined; any nonzero number divided by is undefined. b. 7. = 7. is undefined c. Yes; as expected the quotient is undefined. 11. a. Since the numbers have like signs, the quotient will be positive. b. 8. = 8. =. c. Yes; as expected the quotient is positive. 11. a. Since there are two negative numbers (an even number), the product will be positive. b. (-.)( 4.) (-18) = 6.8 c. Yes; as expected the product is positive False; the product of two numbers with like signs is a positive number 119. False; the quotient of two numbers with unlike signs is a negative number. 11. True 1. False; zero divided by any nonzero number is zero. 1. False; zero divided by 1 is zero. 17. True 19. 4( 1) = -6 The total loss was 6 yards. 11. a. 1 ( 4) = 4 = 1 She paid back $1. b = - Her new balance is $ Ë = 1 - Ë = - 9 or a. It has lost 4 1 points. b. change in October change in September = -41, -, ª.7 The change in October was about.7 times larger than in September. change in October = -41, change in July -, ª 8. The change in October was about 8. times larger than in July. d. You get an error message. 17. a. - = 17 6% of 17 =.6( 17) = 1 7% of 17 =.7( 17) = 17. Target heart rate is 1 to 18 beats per minute. b. Answers will vary. 19

3 19. (-) = (-)( -)(-) = 4( -) = = The product (-1)(- )(-)(-4)L (-1) will be positive because there are an even number (1) of negative numbers. 14. The country will start with D. Most students will select Denmark. They will most likely select kangaroo which leads to orange = 7 1 = 7 1 = 7 or (-18) = = = = -1 - = (- ) = = = 1-7 = (-8) = Exercise Set In the expression a b, a is the base and b is the exponent.. a. Every number has an understood exponent of 1. b. In x y z, has exponent of 1, x has an exponent of, y has an exponent of, and z has an exponent of 1.. a. x + x + x + x + x = x b. x x x x x = x 7. The order of operations are parentheses, exponents, multiplication or division, then addition or subtraction. 9. No; 4 + = = 14, on a scientific calculator. 11. a. = 4 = 1 b. ( ) = = 1 c. The keystrokes in b) are used since the fraction bar is a grouping symbol. 1. b. [ 1 - ( 16 4) ] - 6 = [ 1-4] - 6 = 6-6 = 6-16 = b. When x = : -4x + x - 6 = -4( ) + ( ) - 6 = -4() + () - 6 = = -8-6 = = = = = = -( ) ( ) = -. (-) = (-)(-) = 9. (-1) = (-1)(-1) (-1) = (-9) = (-9)( -9) = (-6) = (-6)(-6) = = 4. (-4) 4 = (-4)(-4)(-4)( -4) = = -( ) ( ) ( ) ( ) = Ë Ë = 4 4 = 9 16 = - 1 Ë - 1 Ë - 1 Ë - 1 Ë - 1 Ë = = = 4. 4 = = a. Positive; a positive number raised to any power is positive. b. 7 = 4 c. Yes; as expected the answer is positive. 47. a. Positive; a positive number raised to any power is positive.

4 b. 4 = 6 c. Yes; as expected the answer is positive. 49. a. Negative; a negative number raised to an odd power is negative. b. (-) = -4 c. Yes; as expected the answer is negative. 1. a. Positive; a negative number raised to an even power is positive. b. (-) 4 = 6 c. Yes; as expected the answer is positive.. a. Positive; a positive number raised to any power is positive. b. ( 4.6) 4 = c. Yes; as expected the answer is positive. 7. a. Negative; is positive therefore, Ë 8-7 is negative. Ë 8 b. - 7 = Ë 8 c. Yes; as expected the answer is negative = + 1 = = + 8 = = 1+ 4 = 1+ 1 = = = 6. ( 4 - ) ( - 1) = ( 1) ( 4) = 1 16 = = 1+ 8 = (7 + ) = - (1) = - 4 = (7 1) = - ( ) = (9) = 4 = Ë = 8 1 = Ë = 4 + = = = = 4-6 = ( 6 ) = ( ) = = 8 + = 1 [ ] = [-8( ) ] (- + ) = (-4) = ( -1) ( + 1) = ( 9-1) ( +1) = 8 (4) = 8 16 = 8 16 = 1 8. [4 + ((- ) ) ] = [4 + (() ) ] = [4 + (9 ) ] = [4 + () ] = [4 + 9] = (1) = ( 9.) = = = = 9.74 [ ] -. = (.7) [ ] -. = (.) -. = = 6.7 1

5 Ë 8 = Ë 4 = = = 4 = = = = 4-4 = = [ ( 6 ) - ] - 4 = = = 49 [ ] = - ( ) [6 - ] = - 8 = = 1 17 { [ ] } = [-- (-) ] ( - ) 1. Substitute for x a. x = = = 9 { } { } { } = [-+ ] = [ ] = -{ 4 - } = -( 4) = -4 b. - x = - = -( ) ( ) = -9 c. (-x) = (-) = (-)( -) = 9 1. Substitute 4 for x a. x = (-4) = (-4)( -4) = 16 b. - x = -(-4) = -(-4)(-4 ) = -( 16) = -16 c. (-x) = 4 = 4 4 = Substitute 6 for x a. x = 6 = 6 6 = 6 b. - x = -6 = -( 6 6) = [ 4 - (6-1) ] [( 9 ) + 4] + = -[4 - (-6) ] ( + 4) ( 6 ) -[4-6] = = -(-) = { [ ] } = { [ ] } { } = - ( 1) = { - (1) } = { - } = ( ) = 9 c. (-x) = (-6) = (-6)(-6) = Substitute - 1 for x a. x = - 1 Ë b. - x = Ë c. ( -x ) = 1 Ë = - 1 Ë - 1 Ë = 1 9 = Ë - 1 Ë = = 1 Ë 1 Ë = Substitute for x in the expression. x + 6 = = 4

6 11. Substitute 4 for z in the expression. z - = ( 4) - = - = Substitute for a in the expression. a - 6 = - ( ) - 6 = 9-6 = 117. Substitute 1 for each x in the expression. -4x - x + 1 = -4(-1) - (-1) + 1 = -4(1) - (-1) + 1 = = = Substitute for each p in the expression. p - 6p - 4 = () - 6() - 4 = (4) = = - 4 = Substitute 1 for each x in the expression. -x - x + = -( 1 ) - ( 1 ) + = = = = Substitute for each x in the expression. 4(x + 1) - 6x = 4(() + 1) - 6() = 4(1+ 1) - = 4(16) - = 4(6) - = 14 - = Substitute for s and 4 for t in the expression. -s + t = -() + (4) = = Substitute for r and for s in the expression. r - s = (-) - (-) = 4-9 = Substitute for x and for y in the expression. (x + y) + 4x - y = [ + (-)] + 4() - (-) = ( + (-6)) (-9) ( ) = -4 = = + 9 = Substitute for x and for y in the expression. 6x + xy - y = 6() + ()(-) - (-) = 6(4) + ()(-) - 9 = 4 + (-18) - 9 = 6-9 = Multiply 6 by ( 6 ) - 4 Subtract 4 from the product ( 6 ) - 4 Evaluate: ( 6 ) - 4 [ ] - = 14 - = 1 [ ] - Subtract from the difference [ ] - = Divide 18 by ( 18 ) + 9 Add 9 to the quotient [( 18 ) + 9] - 8 Subtract 8 from the sum 9{ [( 18 ) + 9] - 8} Multiply the difference by 9 Evaluate: 9 {[(18 ) + 9] } - 8 = 9 {[6 + 9] - 8} 17. = 9[1-8] = 9(7) = Add 4 7 to Multiply the sum by Ë 7 Evaluate: 4 + Ë 7 = Ë = 4 Ë Ë = ( x ) = - x is true for all real numbers When d = 1.99,.7d =.7(1.99) 1.1. The sales tax is $1.1.

7 14. When d = 1,, d +.7d = 1, +.7(1, ) = 1, + 1 = 16, The total cost is $16,. 14. a. = = Ë 7 Ë 14 = - Ë 7 14 Ë - = = (-)() (1)(-) b. ( ) = (.4) = When R = and T = 7,. R +.RT +.1T =.( ) +.( ) ( 7) +.1( 7) =.( 4) +.( ) ( 7) +.1( 49) = = 1.71 The growth is 1.71 inches ( 4-6) + 1 = 4 1. a. There are 4 houses with occupants. b. Occupants Number of Houses c. (1) + () + 4() + 6(4) + () = = = = = 9 There are 9 occupants in all. d. Number of houses = = = = 18+ = number of occupants mean = = 9 number of houses =.9 There is a mean of.9 people per house. Exercise Set 1.1 = 1 or 1 1. The commutative property of addition states that the sum of two numbers is the same regardless of the order in which they are added. One possible example is + 4 = The associative property of addition states that the sum of numbers is the same regardless of the way the numbers are grouped. One possible example is ( + ) + 4 = + ( + 4).. a. In x + y + z ( ) the sum of y and z is added to ( ), x is multiplied by the x whereas in x y + z sum. b. When x = 4, y =, and z = 6, x + y + z ( ) = = 1. ( ) = c. When x = 4, y =, and z = 6, x y + z ( ) = 4( 11) = 44. ( ) = The associative property involves changing parentheses with one operation whereas the distributive property involves distributing a multiplication over an addition a. - 6 b a. b a. - x b. 1 x 16. = 6 = 1 + = Cost = $.4 + (.) = $.4 + $4. = $ a b or (-)(-4)(6)(-1)(-) = (8)(6)(-1)(-) = (48)(-1)(-) = (-48)(-) = a. - 1 b. 4

8 1. a. b. -. Associative property of addition. Distributive property 7. Commutative property of multiplication 9. Associative property of multiplication 1. Distributive property. Identity property for addition. Inverse property for multiplication x 9. (-6 4) 41. x + y 4. y x 4. y+ 4x 47. a + (b + ) 49. x + (4 + 6) 1. (m + n). 4x + 4y n 9. Yes; the order does not affect the outcome so the process is commutative. 61. No; the order affects the outcome, so the process is not commutative. 6. No; the order affects the outcome, so the process is not commutative. 6. Yes; the outcome is not affected by whether you do the first two items first or the last two first, so the process is associative. 67. No; the outcome is affected by whether you do the first two items first or the last two first, so the process is not associative. 69. No; the outcome is affected by whether you do the first two items first or the last two first, so the process is not associative. 71. In ( + 4) + x = x + ( + 4) the ( + 4) is treated as one value. 7. This illustrates the commutative property of addition because the change is + = No; it illustrates the associative property of addition since the grouping is changed = 1 = 1 = 9 1 = = = = 49 1 or = 9 8 = 9 8 = = = = 16 or = = = = 4 8. Substitute for x and for y. -4x + 6xy + y = -4() + 6()(-) + (-) = -4(4) + 6()(-) + (9) = (-6) + 7 = = - Review Exercises 1. 6( 1) - ( ) = 7-64 He had 8 hotdogs left over. [ ] = 1. = (.) [ ] = 1. In years the goods will cost $1..

9 . Less than; the increase of % of the original price is less than the decrease of % of the higher price. [ ] - = ( ) [ ] - = - = a. mean = = 9 = 78.4 The mean grade is ` b. 64, 7, 79, 86, 88 The middle number is 79. The median grade is a. mean = The mean temperature is 8 F. = 4 = 8 b. 76, 79, 79, 8, 84 The middle number is 79. The median temperature is 79 F. 7. a. Profit = 18.% of $4.79 = (.18)(4.79) = 8. The drug manufacturer makes $8. profit on the average U.S. prescription. b. Selling price = Original cost + (Original cost times Markup) = 6 + 6(.) = = 7. The pharmacist will sell the drug for $ a. U.S. oil reserves Canadian oil reserves = = 17.1 The U.S. has 17.1 billions of barrels of oil reserves more than Canada. 9. b. Middle East oil reserves North American oil reserves = ª 1. The Middle East has about 1. times more oil reserves than in North America. c = 18.8 billion barrels The world s oil reserves total 18.8 billion barrels. 6 = = = = 9 1 = 1 9 = 1 9 = 9 1 = 1 = 1 = = = = 7 6 or = = = = = 7 = 7 = = + 1 = 6 = 6 = = = 17 1 or The natural numbers are {1,,, }. 16. The whole numbers are {, 1,,, }. 17. The integers are {,,, 1,, 1,, }. 18. The set of rational numbers is the set of all numbers which can be expressed as the quotient of two integers, denominator not zero. 19. a. and 46 are positive integers. b.,, and 46 are whole numbers. c.,, 1,, and 46 are integers. d.,, 1,, 1,.6, 46, and are rational numbers. e. 7 is an irrational number. f.,, 1,, 1,.6, 7, 46, and are real numbers.. a. 1 is a natural number. b. 1 is a whole number. c. 8 and 9 are negative numbers. d. 8, 9, and 1 are integers. 6

10 e.., 8, 9, 1 1, 1, and - 17 numbers. are rational f.., 8, 9, 1 1,, -, 1, and - 17 are real numbers < ; 7 is to the left of on a number line...6 >.6;.6 is to the right of.6 on a number line... <.9;. is to the left.9 on a number line > 4.6; 4.6 is to the right of 4.6 on a number line... <.;. is to the left of. on a number line. 6. > - since - equals < 7 since -7 equals = since -. = = () = = 1. + (-) = = (-) = -8 + = ( 4) = + 4 =. 4 - (-4) = = = 1 +( 1) = 7. 7 ( 7) = = = + (-7) = (-4) = + 4 = = -7 + (-) = = = 16-9 = = = = = = = = = = = = = = = = = = = Ë 6 = = = + = = - 6 = = -9 - = (-) - = - - = ( +4) - (-) = = + = (-) + = = 6 + = 9 = = 6 = 7 1. Since the numbers have unlike signs, the product is negative; (9) = 7 6. Since the numbers have like signs, the product is positive; ( 8)( ) = 4 7. Since there are an odd number () of negatives the product is negative; (-4)( -)(-6 ) = ( ) (-6) = Ë - Ë 7 = ( -) 7 = -6 = = Ë 11 Ë = (11)(-1) = 6-11 = Ë 8 - Ë 7 = (-)(-) = Zero multiplied by any real number is zero. 4 9 = 7

11 6. Since there are four negative numbers (an even number), the product is positive. (-4)( -6 )(-)( -) = ( 4) (-)( -) = (-48)(-) = Since the numbers have unlike signs, the quotient is negative; 1 (-) = 1 - = Since the numbers have unlike signs, the quotient is negative; 1 (-) = 1 - = Since the numbers have unlike signs, the quotient is negative; - = - = Zero divided by any nonzero number is zero; 4 = 4 = 67. Since the numbers have unlike signs, the quotient is negative; 9 (-9) = 9-9 = = -4 Ë = = -9-1 = = 8 Ë - Ë - Ë - 9 = -6-7 = = 14 Ë = 7 Ë -6 - = -9 = Zero divided by any nonzero number is zero; = = 7. Zero divided by any nonzero number is zero; (-6) = -6 = 7. Any real number divided by zero is undefined; 8 = 8 is undefined. 74. Any real number divided by zero is undefined; -4 = -4 is undefined. 7. Any real number divided by zero is undefined; 8 is undefined 76. Zero divided by any nonzero number is zero; - = 77. ( 8) = -(-) = 78. ( 4-8) = ( -4) = ( - 6) + 4 = = 1 8. (-4 + ) - ( - 6) = (-1) - (-4) = = 81. [ 6 + ( -)] - 6 = [ 6 + (-6) ] - 6 = - 6 = ( 4 )( ) = ( 6)( ) = [ 1 + (-4)] + ( 6-8) = 8 + (-) = [ + (-4)]+ = 9( -1) + = -9 + = -4 [ ] = (-) + 4 (-) ( ) + (-) = (- 4) (- 6) = -1 (-1) = (-)( -4) = = 18 - = [-( ) + 6]- 4 = [-6 + 6] - 4 = - 4 = = (7)(7) = = ( 9) ( 9) ( 9) = = ( ) ( ) ( ) ( ) = (-) = (-)( -)( -) = (-1) 9 = (-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1)(-1 = (-) = (-)( -)(-)(-)(-) = Ë -4 = -4 Ë -4 Ë = 16 = Ë Ë Ë Ë = xxy = x y 98. xyy = xy xxy = 7 x y 8

12 1. xyxyz = x y z 11. x y = xxy 1. xz = xzzz 1. y z = yyyz 14. x y = xxxyy = + = = = 17. ( - 7) + 6 = (-4 ) + 6 = = = 1-9 = 1-7 = = 6-9 = 6-4 = [ 6 - ( ) ] + = [ 6-1] + = -9 + = [9(4 + )] = [9 - (16 + )] = [9-19 ] = (-1) = - = (- + 4 ) + ( ) = ( ) + (9 ) = (7) + () = = = + 18 = 114. (4 ) = () = = = 11. ( 8 - ) = (8-4) = (4) = = = = ( - 7) = (-) = 4 - (-) = 4 - (-) = 4 + = Substitute for x; 6x - 6 = 6( ) - 6 = - 6 = Substitute for x; 6-4x = 6-4( -) = 6 - (-) = 6 + = Substitute 6 for x; x - x + = (6) - (6) + = (6) - + = = 4 + = 4 1. Substitute 1 for y; y + y - = (-1) + (-1) - = (1) - - = - - = - = 11. Substitute for x; - x + x - = - + () - = = - = - 1. Substitute for x; - x + x - = -(-) + (-) - = -4 + (-4) - = -8 - = Substitute 1 for x; -x - x + = -(1) - (1) + = -(1) - + = = -8 + = Substitute for x and for y; - x - 8x - 1y = -(-) - 8(-) - 1(-) = -9 - (-4) + 4 = = = 9 1. a (-49) = -1 b. -49 is greater than 78 so the sum should be (and is) negative. 9

13 16. a. 4 - (-9.6) = = a. b. The sum of two positive numbers is always positive. As expected, the answer is positive = -.88 b. Since the numbers have unlike signs, the quotient is negative, as expected. 18. a. (-6)( -1.9) = b. Since the numbers have like signs, the product is positive, as expected. 19. a. (-) 6 = 79 b. A negative number raised to an even power is positive. As expected, the answer is positive. 1. a. -( 4.) = b. Since ( 4.) is positive, -( 4.) should be (and is) negative. 11. Associative property of addition 1. Commutative property of multiplication 1. Distributive property 14. Commutative property of multiplication 1. Commutative property of addition 16. Associative property of addition Practice Test 1. a. (1.) (1.1) = = = 1.6 The bill is $1.6 before tax. b..7(.). The tax on the soda is $.. c = 1.88 The total bill is $ a. The average cost, per employee, for employers in was about $44.. a. b. $44 - $ = $14 The difference in average cost, per employee, for employers from 1997 to was $14. Population 81 million = Average family size.9 ª 18. million There were about 18. million households in the U.S. in. b. This means half the population was above and half was below this age. 4. a. 4 is a natural number. b. 4 and are whole numbers. c. 6, 4,, 7, and 1 are integers. d. 6, 4, - 1,, 6.,, 7, and 1 are rational 9 numbers. e. is an irrational number. f. 6, 4, - 1,, 6.,,, 7, and 1 are 9 real numbers.. 9 > 1; 9 is to the right of 1 on a number line > - since - = and - = (-8) = = 6 + ( ) = = 17 = (-4 + 6) - ( -) = ( ) - (-6) = + 6 = (-4)( -)( ) (-1) = ( 1) ( ) (-1) = ( 4) (-1) = Ë Ë 8 = = = 16 6 d = 9.1 Her change will be $9.1.

14 = -18 Ë 1 1 Ë = Ë 1 = -9 = = = = (- - ) = -6(-) = = 6 = 1. Substitute 1 for x and for y; - x + xy + y = -(1) + (1)(-) + (-) = -1 + (-) + 4 = = 1. Commutative property of addition 4. Distributive property. Associative property of addition 16. = Ë Ë Ë Ë = ( 11-4) = ( 7) = = 4 4 = 1 { [ ]} = -{-[ ( ) ]} ( 8-6) { } = - -[ ] = -{-[ 16-6] } = -{-[ 1] } = -{-} = 19. Substitute 4 for x; x - 8 = ( -4) - 8 = ( 16) - 8 = 8-8 = 7. Substitute for x and for y; 6x - y + 4 = 6() - (-) + 4 = 6() - (4) + 4 = = = 1 1. Substitute for each x; - x - 6x + = -(-) - 6(-) + = -4 - (-1) + = = 8 + = 11 1

Practice Set 1.1 Algebraic Expressions and Real Numbers. Translate each English phrase into an algebraic expression. Let x represent the number.

Practice Set 1.1 Algebraic Expressions and Real Numbers Translate each English phrase into an algebraic expression. Let x represent the number. 1. A number decreased by seven. 1.. Eighteen more than a