Solutions Key Equations

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1 CHAPTER Solutions Key Equations ARE YOU READY? PAGE 7 1. E. A. C. D. (7 - ) = = = = = 1 = = 1. + (-) = - = = = + 1 = = = = 19. x + = (7) + = 1 + = a = 1 - () = 1 - =. = =. (0.7) + (0.0). 1l. 10 = 0 = = + 0 = (-1) = = (-) = -1 - = = - = - = 1. - = 9 - = 9 - = 0. n - = (7) - = 1 - = 1. y + = () + = 1 + = 0. Possible answer: a number plus times itself -1 SOLVING EQUATIONS BY ADDING OR SUBTRACTING, PAGES 77 CHECK IT OUT! 1a. n -. = n =. b. - = k = k c. 1 = m = m b. - = k = k a m = m = 9. c x = x = a. d + = d = c. + t = t = b. - + z = + + z =. a + r = 0 a + 1 = a = The age of a person who has a maximum heart rate of 1 beats per minute is years old. THINK AND DISCUSS 1. Possible answer: If a scale is balanced, then you can add or remove the same weight on both sides without affecting the balance. Similarly, in an equation, the Addition and Subtraction Properties of Equality say that you can add or subtract the same value on both sides without affecting the equality.. EXERCISES GUIDED PRACTICE 1. Possible answer: The solution of an equation is a number. It is a value for the variable that works in the equation.. s - =. 17 = w s = 1 = w. k - = x -.9 = k = 1 x = 1. Holt McDougal Algebra 1

2 .. = y = y. t + = t = = m = m 1. b + = - - b = d = d = q = + + q = c = + + c = 7. + = t - + = t 9. 9 = s = s 11.. = z = z 1. n + 1. = n = = -1 + v = v 17.. = y = y p = + + p = Let w represent the weight of the original diamond. w - = w = 11 The original diamond weighed 11 carats. Possible answer: The weight of the diamond was reduced by about 0 carats, resulting in a diamond that weighed about 70 carats. So the original weight will be close to = 10 carats. So 11 carats is reasonable. PRACTICE AND PROBLEM SOLVING 1. 1 = k -. u - 1 = = k u = 7. x - 7 = = p x = = p. 7 = p -. q - 0. = q = = p 7. = t -.. = r = t + + = r 9. = x = x a = a = 0. m + 0 = m = -17. v + 00 = v = = n = n 9. x +. = x = f = f = j = + + j = 1. = - + y = y h = + + h = = k = k. =.1 + y = y. -1 = c = c. b + = b =. b + = - - b = 0 0. = d = d. -9 = - + g = g. 90 = - + a = a. 1 = - + w + + = w a = a = Let a represent the amount in Luis s acount before the deposit. a + 00 = a = Luis had $ in his account before the deposit. Possible answer: Luis deposited $00 and now has about $700. So the will be close to = 00. So $ is a reasonable answer. 0. Solution B is incorrect. The Addition Property of Equality was used. 1. x - 10 = x =. x + = x =. x - 1 = x = 0. x - = x = - Holt McDougal Algebra 1

3 . + x = - - x = 1. x - = x = - 7. x - = x = 1. Let a represent the average depth of the Atlantic Ocean. a + 0, =,1-0, - 0, a = 1,0 The average depth of the Atlantic Ocean is 1,0 ft. Possible answer: The greatest depth is about 0,000 ft. The sum of this number and the average is about,000 ft. So the average will be close to,000-0,000 = 1,000. So 1,00 ft is a reasonable answer. 9. Let m represent the amount of money the band still needs. m + 0 = m = 110 Helene s band still needs to raise $110. Possible answer: The band needs about $1700 and they have raised about $00. So the amount they need will be close to = 1. So $110 is a reasonable answer. 0. Let c represent the cost of the car. c - = c = The car cost $. Possible answer: The total cost should be more than the down payment and more than the loan amount, so $ is a reasonable answer x = x = x = 7 x =. x + 1 = x = 7 a. 000 acres; possible answer: The fire should cover twice as much area in days as it does in 1, so multiply by 0 and the answer is 000 acres. b. 000 acres; multiply days by 0 acres per day. c. Divide 70 by 7.. Let h represent the highest score. h - 7 = h = 7 The highest score is 7. Possible answer: The range is about 0 and the lowest score is about 0. So the highest score will be close to = 0. So 7 is a reasonable answer.. Possible answer: If you add years to Sue s age, you get her cousin s age. Her cousin is. How old is Sue? x represents Sue s age. x + = - - x = 0 Sue is 0 years old. 7. Possible answer: greater than 10 because you will add a positive number to both sides TEST PREP. A: is being subtracted from x so the situation must decrease by. Choice A has a withdrawal of. 9. J: Since + = 1, J is an equation for which a = is a solution. 70a. g - 1 = b. $0 g - 1 = g = 0 CHALLENGE AND EXTEND b = - - b = x + 7 = x = p - = + + p = p - 0 = () - 0 = 0-0 = x + = x = x = 1 + (1) = = 90 SPIRAL REVIEW (-7) = 9 1. (-1)(-) = 7 7. x - 7 = x = x - 9 = x = 7. t + = t = 1 -t = -(1) = n = n = n = (-1) = -7 7) 7 ( - = = -. = 1 = 1 m. = = ft. = 10 = 10 cm 7 Holt McDougal Algebra 1

4 . [- - ( + )] = (- - ) = (-10) = (-) 7. = -1 = [ - (1-1) ] = 1- [ - (-) ] = 1 - (1-9) = 1-7 = - - SOLVING EQUATIONS BY MULTIPLYING OR DIVIDING, PAGES 90 CHECK IT OUT! 1a. c. p = 10 () ( p ) = ()(10) p = 0 c = 7 () ( ) c = ()(7) c = b. 0.y = y 0. = y = -0 a. - = b c. () ( - ) = () ( - = b b ) w = 10 () ( w ) = ()(10) d w = 1 b. -1 = y ()(-1) = () ( y ) -9 = y a. 1 = c 1 = c = c c. 1k = 7 1k 1 = 7 1 k = j b. = ( ) j = ( ) j = 1. = h = h 1 = h The plane was 1,000 ft high when it began its descent. THINK AND DISCUSS 1. Possible answer: All four properties tell you that if you perform the same operation on both sides of the equation, the equality will still hold.. EXERCISES GUIDED PRACTICE 1. k = () ( k ) = ()() k =. - = w -7 (-7)(-) = (-7) ( -7) w 1 = w. g 1.9 = 10 (1.9) ( 1.9) g = (1.9)(10) g = x = x = x = j = - -9j -9 = - -9 j = 11. m = 10 m = 10 m = d = 7 () ( d ) = ()(7) ( d = 1 s = - ) (-) s = -9 ) ( s ) = ( 10 = ( ) y ( ) 10 = ( ) = y y. z = -9 () ( z ) = ()(-9) z = -7 t - (-)() = (-) ( t -) -0 = t. =.. = b ()(.) = () ( b ) 1 = b. - = c - = c - = c 10. = -1a -1 = -1a -1-7 = a 1.. = -h. - = -h = h 1. 1 = f ( ) (1) = ( ) ( 1 = f 1. 9 = - r 1. ( Let c represent the cost per child. 1c = 19 1c 1 = 19 1 c = 1 The cost per child is $1. f ) ) (9) = ( - ) ( - - = r v = - () ( v ) = () ( - ) v = - r ) 0. Let a represent the amount of vitamin C in an apple. 0 = 10a 0 10 = 10a 10 = a There are mg of vitamin C in an apple. Holt McDougal Algebra 1

5 PRACTICE AND PROBLEM SOLVING 1. x = 1 () ( x ) = ()(1). - x = j = (-) ( - j ) = (-)() j = -. - q = 0 (-) ( - q ) = (-)(0) 7. q = -10 v 10 =. (10) ( v 10) = (10)(.) 9. t = -1 t = -1 t = - v = = -1u -1-1 = -1u -1 1 = u. - = -c - - = -c - 1 = c. f = 1. f = 1. f = k = ( ) ( k ) = ( ) () k = 9. - b = 10 ( - ) ( - b ) = ( - ) (10) b = g = -1 ( - 1. ( 7 ) ( - g ) = ( - 7 t = - ) ( 7 t ) = ( = b ()(-0) = () ( b ) -00 = b. - n = - (-) ( - n ) = (-)(-) n = = d ()(1.) = () ( d ). = d. ) (-1) g = 1 ) (-) t = -. h.1 = - (.1) ( h.1) = (.1)(-) 0. 9 = 7c 9 7 = 7c 7 7 = c. -7m = -7m -7 = -7 m = = -z 1 - = -z - -. = z h = = -n -. - = -n -.1 = n. -9 = d ( ) (-9) = ( ) ( ) d -1 = d. - p = ( - ) ( - p ) = ( - ) ( ) p = -. (-) ( = - q ) = (-) ( - - = q. - = - a ( - q ) ) ( - ) = ( - ) ( - = a a ). Let s represent Alexandra s salary before taxes s = 9 ( 10 7 )( 7 7 ) (9) s = 0 Alexandra s salary before taxes is $0. 10 s ) = ( 10. Let w represent the person s weight on Earth. w = 1 () ( w ) = ()(1) w = 9 The person weighs 9 lb on Earth. Possible answer: The person s weight on the Moon is about 1 lb. The answer will be close to 1 = 90. So 9 lb is reasonable. 7. Possible answer: The student divided both sides by instead of multiplying both sides by. x = 1 () ( x ) = ()(1) x =. s = s = s = 9 in. 0. s = s = s = yd. x = x = x = 9. x = 10 () ( x ) = ()(10) x = 0 9. s = s = s = 1 in. 1. s = 1. s = 1. s =.1 cm. -x = 1 -x - = 1 - x = -. x = - () ( x ) = ()(-) x = -. Let x represent the total measure of all students heights. x 1 = 0 (1) ( x 1) = (1)(0) x = 100 The total measure of all students heights is 100 in. 9 Holt McDougal Algebra 1

6 7. Let h represent the number of hours Lisa worked each week..h = 0.h. = 0. h = Lisa worked hours each week.. Greater than ; you are multiplying by a positive number greater than Let m represent the number of minutes Dion was charged for. 0.0m = m 0.0 = m = 7 Dion was charged for 7 minutes. Possible answer: Dion s calls cost $0.0/min. If he talked for 00 min, the cost would be 00(0.0) = $1. This is close to the cost given in the problem, $1.0. So 7 min is reasonable. 0. Let c represent the amount of caffeine in a 1 oz caffeinated soft drink. c = 1 c = 1 c 7 There are about 7 mg of caffeine in a 1 oz caffeinated soft drink. Possible answer: The amount of caffeine in the soft drink should be about 00 = 0 mg. So 7 mg is reasonable. 1. y = x y = (-) y = -1 y = -1 y = -. y = x y = (0) y = 0 y = 0 y = 0. y = x y = (-) y = - y = - y = -1. y = x y = () y = y = y = 1 a. number of data values b. mean c. mean = sum of data values number of data values 9.1 = x 19 (19)(9.1) = (19) ( x 19) 1,00 acres x Possible answer: There were about 1900 fires and the mean acreage burned was about. So the total acres burned will be close to 1900() = 190,000. So 1,00 is reasonable.. m = 1 () ( m ) = ()(1) m =. 1.h = 1. 1.h 1. = h = w = w = w = 1 7. y = 11 y = 1 y = 1 7. x = x = x = 7 9. x = 11 () ( x ) = ()(11) x = b = 7. ( ) (b) = ( ) ( ) b = 1 n 1.9 = (1.9) ( n 1.9) = (1.9)() n =.7 7. Let m represent the mean weight of an adult male mouse. 1m = 0 1m 1 = 0 1 m = 0 The mean weight of an adult male mouse is 0 g. Possible answer: An adult male rat weighs about 00 g, and this is about 0 times the weight of an adult mouse. So the mouse will weigh close to 00 = g. So 0 g is reasonable Let g represent the average weight of a gerbil at birth. g = ( ) () g = The average weight of a gerbil at birth is g. ) ( g ) = ( Possible answer: A hamster weighs g at birth, so the gerbil s weight will be greater than. So g is reasonable. 7. Possible answer: Ryan has times as many baseball cards as David. Ryan has cards. How many cards does David have? x = x = x = 1 This represents the number of cards that David has. TEST PREP 77. D; Since situation D says Mattie had more, the situation involves addition. Since the given equation does not contain addition, D cannot represent the equation. 0 Holt McDougal Algebra 1

7 7. J; Since x is multiplied by, the first step is to divide both sides by. Therefore, equation J is correct. 79. B; There are 0 in a pentagon and angles. If x is the size of each angle, then angles must add to 0. Therefore x = 0 is correct. 0. H; Since 10 =, m = 10 is a solution for equation H. 1a. c =.0; is the number of cans of cat food; c is the cost per can; $.0 is the total cost. b. $0.0 c =.0 c =.0 c = 0.0 One can of cat food costs $0.0. CHALLENGE AND EXTEND. ( ) b = 1 b = 1) 1 b = ( 1) b = x = - ( ). (. ( 1 ) x = x = ( ) x = ( ) x = 9 x = - 1 ( 9) 9 x = ( 9) ( - 1 ) x = -9. (- 9 k = - 10) k = ( ) ( k ) = ( ) ( ) k = 9. ( 1 w = 1 7. ) ( ) d = w = 9 ( ) w = ( ) d = 9 ( 9) 9 w = d = ( 9) 9 d =. p = p = p = p + 10 = () + 10 = = 9. t = t = t = -t = -() = x = 1 x = 1 x = 1 - x = 1 - () = 1-0 = n = -11 () ( n ) = (-11) n = - n = (-) = a 9. Multiply both sides by a. 9a. d = rt 00 = t 00 = t 1 = t b. d = rt c. t is divided in half (from 1 to ). d. t is divided in half. SPIRAL REVIEW 00 = 0t 00 0 = 0t 0 = t 9. 1 = 1 = = 1 = = = = - = Let l represnt Lisa s age. l + 11 = l = Lisa is years old.. Let w represent the width of the rectangle. = w = w The width of the rectangle is ft = a = a 10. z - = z = x - 1 = x = = x = x - SOLVING TWO-STEP AND MULTI-STEP EQUATIONS, PAGES 9 9 CHECK IT OUT! 1a x = + + 7x = 7 7x 7 = 7 7 x = 1 b. 1. = 1.y = 1.y = 1.y 1. = y 1 Holt McDougal Algebra 1

8 c. n 7 + = - - n 7 (7) ( n b. u + = 0 7) = (7)(0) n = 0 - = 7 - u = ( ) u = ( u = ) a. a + - a = a - a + = -a + = - - -a = -a - = - a = - b. -( - d) = (-)() + (-)(-d) = - + d = + + d = 10 d = 10 d = c. (x - ) + x = 0 ()(x) + ()(-) + x = 0 x - + x = 0 x + x - = 0 x - = x = x = x = a. x - = + + x = 1 ( ) x = ( ) 1 x = = c. n - +. Let x represent Sara s monthly fee. 1x = x = x 1 = x = 0.00 Sara s monthly fee was $ n = () ( n ) = ()() n = 1. x + = x = - x = - x = -1 x = (-1) = - THINK AND DISCUSS 1. Possible answer: To solve x + 1 = 7, first subtract 1 from both sides and then divide both sides by. To solve x - 1 = 7, first add 1 to both sides and then divide both sides by.. EXERCISES GUIDED PRACTICE 1. a + = a = a = a =. = -d = -d - = -d - -1 = d. 1y + 1 = y = 0 1y 1 = 0 1 y = 7. x + = = 11 x ( x ) = (11) x =. = r = r 9 = r = r. x + 0. = x =. 9 - c = c = - (-1)(-c) = (-1)(-) c =. y + - = 1 - y = ( y ) = ( ) y = Holt McDougal Algebra 1

9 9. 7 j - 7 = j = 1 ( 7 ) 7 j = ( 7 j = ) m = m = (-) ( - m ) = (-)() m = = x + 1-7x = x - 7x + 1 = x = x 1.. = (m + ). = ()(m) + ()(). = m = m -9. = m -. = m 17. t t = 19 t - t + 7 = 19 t + 7 = t = 1 t = 1 t = 1. (1 - w) + w = 1 ()(1) + ()(-w) + w = 1-10w + w = 1 - w = w = 10 -w - = 10 - w = = a = a (17) = ( a ) 1 = a 1. x - = + + x = 1 ( ) x = ( 1 ) x = 1. y y = 0 y + y - 7 = 0 7y - 7 = y = 7 7y 7 = 7 7 y = 1 1. (x - ) = ()(x) + ()(-) = x - 1 = x = 0 x = 0 x = Let x represent the number of daily bus passes Paul bought. 1.0x + 7 = x =.0 1.0x 1.0 = x = 1 Paul bought 1 daily bus passes. 0. x - 1 = x = 1 x = 1 x = 7 1. (x + 1) = 7 ()(x) + ()(1) = 7 x + = x =. -(y - 1) = 9 (-)(y) + (-)(-1) = 9 -y + = y = -y - = x = 9 y = x = -7x -7 = -7 x = - x - = 7 - = y = (-) = -1 x + 1 = = - PRACTICE AND PROBLEM SOLVING. = g + 1. h - 7 = = g h = = g h = = g h =. 0.v +.1 =. 7. x + = v =. 0.v 0. = x = 1. x 0. = 1 v = x =. 0.g + 11 = 9. = - t g = - 7 = -t 0.g 0. = = - g = = t Holt McDougal Algebra 1

10 0. d + =. z. - - d = (d) = ( d = + 1 = z ( z + = ) = ( ) ) z = 1 = x - + ( ) 9 9 = x = ( = x ) x 1. 1 = x +... = -(7 - c) = (-)(7) + (-)(-c) = -1 + c = c 0 = c 10 = c 7. (h - ) = ()(h) + ()(-) = h - 0 = h = h = h =. -x - + x = 17 -x + x - = 17 x - = x = 9. x + x = 0 10x = 0 10x 10 = 0 10 x = - - = x ( ) = = x = j (x) ( ) = ( ) j - 1 = j - x = x = - (-) ( - x ) = (-) ( - ) x = 0. (x + ) = 10 ()(x) + ()() = 10 x + = x = x = x = = (p - ) + 17 = ()(p) + ()(-) + 17 = p = p = p = p = p. Let w represent the number of weeks. = 1 + 1w = 1w 10 1 = 1w 1 = w Jennifer needs to save for another weeks to buy the bike.. x + 1 = x = x = x = x + 1 = () + 1 = + 1 = 7. -(x - 1) = (-1)(x)+(-1)(-1) = -x + 1 = x = (-1)(-x) = (-1)() x = - -x = -(-) = 1. (y+10) = x = ()(y) + ()(10) = y + 0 = 0 -x = x y = -10 y = - = x = - x - y = - = - - = -10 y = (-) = - 7. (x + 7) = 10 x + = x = 0 x = 0 x = 0 Holt McDougal Algebra 1

11 . x + x + 11 = 10 x + 11 = x = x = x =. 9. (x - 0) = 10 x + 0 = x = 10 x = 10 x = 0. n - 7 = n = n = n = 1. n + = n = n = n = 1. - n = - - -n = - -n - = - n = a. Let s represent the number of years in a score s = s = - -s - = - - s = 0 There are 0 years in a score. b. Let n repesent the number of scores. ns = 0 n(0) = 0 0n 0 = 0 0 n = There are scores in 0 years.. t + = t = t = t =. 1 = c = c (17) = ( c ) 1 = c. (x - ) = 1 ()(x) + ()(-) = 1 x - = x = x = x = 7. x +. = x = 9 x = 9 x =...9w = w = 1..9w.9 = 1..9 w = = x - (x + 1) 17 = x + (-)(x) + (-)(1) 17 = x - x - 17 = -x = -x 0 - = -x = x 0. x + 9 = m = x = 0 x =.m = 0.m. =. x = m = 10. Let k represent the height of a kiwi. k + 0 = k = k = k = A kiwi is in. tall. Possible answer: The height of an ostrich will be more than times the height of a kiwi. 10 is more than () =, so in. is reasonable.. Let k represent the height of a kakapo. k - 70 = k = 10 k = 10 k = A kakapo is in. tall. Possible answer: The height of a kakapo will be more than the height of an emu. is more than (0) = 1, so it is a reasonable answer.. Let n represent the first number. (n) + (n + 1) = 7 n + 1 = n = n = n = The numbers are n = and n + 1 = 9. Holt McDougal Algebra 1

12 . Let n represent Stan s age. (n) + (n + 1) + (n + ) = 111 n + = n = 10 n = 10 n = Stan is n =, Mark is n + 1 = 7, and Wayne is n + =.. Let n represent the first number. (n) + (n + ) = 0 n + = n = 0 n = 0 n = 10 The numbers are n = 10 and n + =10. 7a. b. c = n Cost of Fighting Fire Acres Cost ($),00 00, ,00 0,000 7,00 n n. Possible answer: Distribute or subtract from both sides. 9. Possible answer: Simplify both sides if necessary. Find the term containing the variable. Undo any addition or subtraction on this term by using inverse operations. Then undo any multiplication or division on this term by using inverse operations. TEST PREP 70. D; g represents the number of shirts Greg sold. Lin sold g + and Fran sold (g + ) = g + 1. Therefore, the total number of shirts sold is g + g + + g + 1 = 1 + g, so D is correct. 71. H; Solving for m reveals m =, so 7m - is - = C; Since m is multiplied by 0.0, there is a cost of $0.0 for each mile. Since is added to 0.0m, there is an additional fee of $. Therefore C is correct sales 10 + s = s = s = s = 7 CHALLENGE AND EXTEND 7. 9 x x = 1 9 x + x + 1 = 1 1 x + 1 = x = - ( 1) ( 1 x ) = ( 1) ( - ) x = x - 1 = x = 9 ( 1) 1 x = ( 1) 9 x = 7. (x + ) - (x + 7) - x = -9 x + + (-1)(x) + (-1)(7) - x = -9 x + -x -7 - x = -9 x - x - x = -9 -x - 1 = x = - -x - = - - x = 77. (x + ) - (1x + ) + (x - ) = + 11 x + - 1x x - = + 11 x - 1x + 10x = + 11 x - 1 = x = 9 x = 9 x = x + b = -1 () + b = -1 + b = b = -9 b = -9 b = - 0a. p = nc - e 00 = 000c = 000c = 000c = c c. c doubles 79. x - b = 0 (-9) - b = b = b = 1 -b - = 1 - b = - b. p = nc - e 00 = 0c = 0c 00 0 = 0c 0. = c Holt McDougal Algebra 1

13 SPIRAL REVIEW 1. irrational. integer, rational, terminating decimal. repeating decimal, rational. terminating decimal, rational. (1) = (0) + (1) = 0 + = 7. 11() = 11(0) + 11() = 0 + = = k = k 91. a + = a = -1. 9() = 9(0) + 9() = =. 1(1) = 1(0) + 1(1) = = x - 1 = x = = q = q - SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES, PAGES 10 CHECK IT OUT! 1a. b + = b - b - b b + = b = - b y = 0.7y y - 0.y 0. = 0.y = 0.y = 0.y 0. = y a. (b + ) = b - 1 (b) + () = b - 1 b + = b b b = b = b b. x = (x + ) x = (x) + () x = x + x + = x + - x - x x + = - - x = - a. y y = 10 + y 7 + y = 10 + y - y - y 7 = 10 no solution b. c c = -1 + c + 1 c + 7 = c c - c 7 = 7 all real numbers. Let g represent Greg s age. g - = g g - g g - = g = 10 Greg is 10 years old. THINK AND DISCUSS 1. Equation b,.x x = + 9x - 11 is an identity. The equation simplifies to 9x - 9 = 9x - 9 which is true for all values of x.. EXERCISES GUIDED PRACTICE 1. Possible answer: After simplifying, the. c - = c + - c - c expressions on either c - = side of the equal sign + + are the same. c = 9. r + = 10 + r. x - 1 = x r - r - x - x r + = 10 x - 1 = r = r = x = 1 r = y = 0.7y y + 0.y = y = y 7 Holt McDougal Algebra 1

14 . -(x + ) = x - -(x) - () = x - -x - = x - + x + x - = x = x - = x - = x 7. c - c + 1 = c + + -c + 1 = c + + c + c 1 = c = c - = c - = c. + (q - ) = (q + 1) + (q) + (-) = (q) + (1) + q -1 = q + q - 7 = q + - q - q q - 7 = q = (t + ) = -1 + (t - ) - (t) - () = -1 + (t) + (-) - t - = -1 + t - - t = t t + t = t = t 9 = t = t 10. 7x - = -x x - 7x - = 7x - - 7x -7x - = - all real numbers 11. x + - 9x = - x x = -1 - x + x + x = -1 no solution 1. y = y y = -1 + y - y - y 0 = -1 no solution 1. - x - 1 = x + - x - - x = - x + x + x = all real numbers 1a. Let x represent the number of hours x = 0 + 1x - 1x - 1x 7 = 0 + x = x 9 = x = x The two companies cost the same amount for a -hour job. b x = 7 + 1() = 7 + = $70 PRACTICE AND PROBLEM SOLVING 1. 7a - 17 = a a - a a - 17 = a = 1 a = 1 a = 17. x - = x + - x - x x - = + + x = 1. b - = b b - b - = b = b - = b -1 = b 1. x - = x x - x - = x = x - = x - = x 19. x - = x x - x x - = x = 1. x = 1. x =. 0. x + = x 1. c - = c + - x - x - c - c = -x - = c - = -x c = 10-1 = x x = - x. (t - 1) = 9 + t + x + x (t) + (-1) = 9 + t -17 = + x t - = 9 + t t - t - = x t - = t = 1 t = 1 t = Holt McDougal Algebra 1

15 . - x - = + x + - x = x + + x + x = x = x - = x -1 = x. (x + ) = (x - ) (x) + () = (x) + ( -) x + = x - - x - x = x = x. m - 10 = (m - ) m - 10 = (m) + (-) m - 10 = m m - m -10 = m = m 0 = m 0 = m 7. - (n - ) = (n + ) - 1(n) - 1(-) = (n) + () - n + = n n = n + + n + n 9 = n = n = n = n. (x + 7) - 0 = x (x) + (7) - 0 = x x = x x + = x - x - x = 0 no solution 9. (x + 1) = x - (x) + (1) = x - x + = x - - x - x x + = x = -1 x = x = - 0. x - - x = -x x - = -x - + x + x - = - all real numbers 1. -(x + ) = -x + 1 -(x) - () = -x + 1 -x - = -x x + x - = 1 no solution. (x + ) - = x + (x) + () - = x + x + - = x + x + = x + + x + x = all real numbers a. Let w represent the number of weeks w = 19 - w + w + w 10 + w = w = w = w = 1 Justin and Tyson will weigh the same amount in 1 weeks. b w = 19-1 = 10 lb. (x + ) = 1 + x (x) + () = 1 + x x + 1 = 1 + x - x - x x + 1 = x = x = x =. x - 0 = 1 - x. x - = x + + x + x - x - x x - 0 = 1 x - = x = x = x = x = x - = x + - x - x - = x = x - = x - = x. x + = x + - x - x x + = - - x = - 9 Holt McDougal Algebra 1

16 9. x + = x - - x - x = x = x 0. - p + = p p p ( = p = p ) ( p ) -1 = p ) (-9) = ( 1. x + = x + 1. x - 10 = 1 - x - x - x + x + x x + = 1 x - 10 = x = -9 x = x = -9 x = x = - x =. 1 - x = 10 - x. x - 7 = -x x + x + x + x 1 = 10 + x 11x - 7 = = x 11x = - 11x 11 = 11 x = -. 1.x +. =.x x - 1.x. = 0.7x = 0.7x = 0.7x = x..x + 1 =.x + -.x -.x 0.x + 1 = x = 0.x 0. = 0. x = x = x +. + x + x 1 = x = x -7 = x - 7 = x 9. (x + 1) = x + 7 (x) + (1) = x + 7 x + = x x - x x + = x = ( - h) = h () + (-h) = h - h = h + h + h = h = h 1 = h 0. 9x - + x = 7x + 1 1x - = 7x + 1-7x - 7x x - = x = x = x = 1. (x - 1) + = (x + 1) (x) +(-1) + = (x) + (1) x - + = x + x + = x + = There is no solution. a. Let m represent the number of miles m = + 0.m + 0.m = + 0.m - 0.m - 0.m = m = 0.10m = 0.10m 0.10 = m + 0.m = + 0.() = + = $0 Both companies charge $0 for a -mile rental. Possible answer: For mi, Rapid Rental Car charges $0 + $1 + $ = $0. Capital Cars charges $ + $ = $0. b. Capital Cars; the cost would be less. c. No; now Rapid Rental Car would be cheaper. d. Less than mi - use Capital Cars; more than mi - use Rapid Rental Car.. (x + ) = (x + ) + (x - ) + x (x) + () = (x + ) + (x - ) + x x + 9 = x + + x - + x x + 9 = x - x - x 9 = x a () = = 0 acres Possible answer: The total acreage in the next days will be about more than 0, so 0 is reasonable. b d c. 0d d d = 0d - 0d - 0d 0 = 0d 0 0 = 0d 0 1 days = d d represents the number of days it will take the firefighters to put out the fire.. Possible answer: x + = x + + x 0 Holt McDougal Algebra 1

17 a. 7x b. x c. 7x + 1 d. 7x + 1 = x - 7x - 7x 1 = 1x 1 = 1x 1 1 = x The cheetah will have to run for h or about 1 min to catch the gazelle. e. No; the cheetah can only maintain its top speed for about 0.00 h, or about 11 s. 7. Possible answer: Simplify both sides if necessary. Collect the variable terms on one side by using inverse operations. Then isolate the variable using inverse operations. TEST PREP. C; Lindsey s monthly magazine costs $1. per issue so her total cost is 1.m. Kenzie got her first two magazines free so she paid for m - magazines. Since Kenzie s magazines cost $1.0, her total cost is 1.0(m - ). So when 1.m = 1.0(m - ), they will have paid the same amount. 9. F; The equation is 7x = x -. Solving gives x = -1., which is F. 0. B; packs of markers plus $9.00 will be equal to the cost of packs of markers so B is correct. 1. H; In 1 days, or 19 weeks, she will save an additional 0(19) = $0. So she will have = $00.. -(x - 1) + x = (x - 1) -(x) - (-1) + x = (x) + (-1) -x + + x = x - x + = x - - x - x = x = x CHALLENGE AND EXTEND. x + [ - (x + )] = x - x + () + [-(x + )] = x - x + - (x + ) = x - x + - (x) - () = x - x + - x - = x - 0 = x = x = x = x.. x + + x - 1 = ( x + + x - 1 ( x + - x - 1 ) ) ) = ( x - 1 ) + ( x - 1 ) = ( x - 1 x x - = x - x + 1 = x - - x - x x + 1 = x = -1 x = -1 x = - 7 w - = ( w - ) w - = (w) + ( - ) w - = w - - w w - = - no solution f = -f -(f + ) f = -f - (f ) - () f = -f -f f = -f f + f -1 = -1 all real numbers 7. x + = x -. x - = x x x = - 1 x x = - (1) ( 1 x ) = (1) ( - ) x = x x x - = x = ) ( x ) = ( ) () x = 1 9. Let n represent the first number. (n + ) = n - (n) + () = n - n + = n - - n - n = n = n The numbers are n =, n + 1 = 7, and n + =. ( 1 Holt McDougal Algebra 1

18 70. Let n represent the first number. n = 1 + n + n = 1 + n -n - n n = 1 The numbers are n = 1, n + 1 = 1, and n + = Let r represent the amount of money Rob originally had. r = r - 0. = r r - r r - 0. = r = 0. () ( r ) = ()(0.) r r = 1. Rob originally had $1.. SPIRAL REVIEW 7. x cm 7. y cm 7. undefined 7. (-1) = = 0 10 = 7 = 10 0 = 10 7 = 0 = 7 = 0 7. (-) = - = (-0.001) = -0 ( - 1 ) = -0 - = 1,000,000. x - = + + x = x = x = = x = x ()() = () ( x ) = x = 1 = 1. 00(-0.) ) = 00 ( - = -00 = -1. (x - ) = (x) + (-) = x - = + + x = 0 x = 0 x = 1. x + = x = x = x = - SOLVING FOR A VARIABLE, PAGES CHECK IT OUT! 1. d = rt d r = rt r d r = t t = d r t =. 1 t 1. It would take Van Dyk 1. hours. a. - b = t - b = t - b = t. f = i - gt + gt + gt f + gt = i b. D = m V (V)(D) = (V) ( m V ) VD = m VD D = m D V = m D THINK AND DISCUSS 1. Possible answer: The formula d = rt is more useful in the form d = t when you need to determine the r time it takes to travel a certain distance at a certain speed.. Possible answer: Isolate w on the right side by subtracting l from both sides and then dividing both sides by.. EXERCISES GUIDED PRACTICE 1. A literal equation contains more than one variable. A formula shows how to determine the value of one variable when you know the value(s) of one or more other variables. So a formula always contains more than one variable, making it a literal equation. a. a = c a = b. c = a c c = a = c c = 7 The room would need 7 circuits. Holt McDougal Algebra 1

19 . V = lwh V lh = lwh lh V lh = w. m - n = + n + n m = + n 7. b + c = 10 a (a)(b + c) = (a)( 10 a ) a(b + c) = 10 a(b + c) (b + c) = 10 (b + c) a = 10 b + c PRACTICE AND PROBLEM SOLVING a. C = πr C π = πr π C π = r 9. A = P + I - P - P A - P = I 11. xy - = k + + xy = k + xy y = k + y x = k + y 1. x - = z y (y) ( x - y ) = (y)(z) x - = yz x - = yz z z x - = y z 1. x - g = a + g + g x = a + g () ( x = ()(a + g) ) x = (a + g). st + t = - t - t st = - t st = - t t t s = - t t f +. = g (g)( f + g ) = (g)() f + = g - - f = g - b. r = C π r = π r = 7. π in = r + s - r - r - - r = s 1. m n = p - (n) ( m n ) = (n)(p - ) m = n(p - ) m n(p - ) = (p - ) (p - ) m p - = n 1. S = 10n S + 0 = 10n S + 0 = 10n S + 0 = n A = bh A = h b ( h) (A) = ( h) ( h A h = b b ) 17. y = mx + b - b - b y - b = mx y - b m = mx m y - b m = x 19. PV = nrt PV nr = nrt nr PV nr = T 1. M = T - R + R + R M + R = T. a + b = c - a - a b = c - a b = c - a b = c - a 1. a = n a - 1 = n a - 1 = n a - 1 = n 0. T + M = R - M - M T = R - M. PV = nrt PV nt = nrt nt PV nt = R. p + 9c = p - p - p 9c = -p 9c 9 = -p 9 c = - p 9. ax + r = 7. x + 7y = - ax - ax - x - x r = 7 - ax 7y = - x 7y 7 = - x 7 y = - x 7 7. y + x =. y = x + b - y - y - x - x x = - y y - x = b x = - y y - x = b x = - y y - x = b 9a. Possible answer: t d 00 b. Possible answer: t d 00 = 00 =. It takes about. h to fly mi. c. Possible answer: t d 00 (00)(t) (00) ( d 00) 00t d d. Possible answer: d 00t = 00() = 000 The airplane can fly about 000 mi in h. 0. Ei = 9r Ei = 9r i i E = 9r i E = 9() 1 E =. The pitcher s ERA is.. Holt McDougal Algebra 1

20 1. t = -0.00a + g - g - g t - g = -0.00a t - g = -0.00a t - g = a. Possible answer: Use inverse operations to isolate the indicated variable on one side of the equation. You must be sure the variable is the only expression on one side of the equation and doesn t appear on the other side.. Possible answer: The variable a appears in terms. Use the Distributive Property to write a - ab as a(1 - b), and then divide both sides by 1 - b. a. Days Acres d 0d b. A = 0d c. The graph is a line. TEST PREP. C; First, subtract 9 from both sides. Then divide both sides by. Simplyfing the fraction gives C.. J; To isolate the b variable on the left side, subtract a from both sides. Then, since b is multiplied by -, divide both sides by - to undo the multiplication. 7. D; Since Anna knows the length, width, and volume of the box, she need to find the height and therefore needs to solve for h. CHALLENGE AND EXTEND..x + r =.1 - r - r.x =.1 - r.x. =.1 - r. x =.1 - r. 9. a - b = c + b + b a = c + b ( 0. ) ( a ) = ( ) ( c + - b ) a = ( c + b ) x + 1.y = - x x 1.y = - x 1.y 1. = - x 1. y = - x t = - t - (00) ( t - 00t - 0 = d. s = g t ( s = t g ) (s) = t ( ) ( t t d = d 00 00) ) = (00) ( d g) s = g t. y = mx y - = mx y - = x mx x y - = m x If m = 0, y - = 0. Therefore y =.. S = hwft,000 (,000 hwf,000s = t hwf t =,000S hwf t =,000(70) 0(1)(1) t 10 s hwf ) ( (S) =,000 )( hwft. v = u + as - u - u v - u = as v - u = as a a v - u = s a,000) Holt McDougal Algebra 1

21 SPIRAL REVIEW. (0) = $10 7. (0) = 1 girls. + x + = + () + = + + = x = = -. = - + c + + = c a = a = x = - 1 = - 1 = = - + w = w. - + k = + + k = 1 - SOLVING ABSOLUTE-VALUE EQUATIONS, PAGES CHECK IT OUT! 1a. x - = + + x = 7 Case 1 Case x = 7 x = -7 x - = x - = b. = x -. Case 1 = x = x Case - = x = x = x -. = x a. - x - = x - = (-1)(- x - ) = (-1)() x - = - no solution b. - + x - = x - = 0 x - = x =. First convert millimeters to meters: 10 mm = 0.1 m. Find two numbers that are 0.1 units away from 1 by using the equation x - 1 = 0.1. Case 1 x - 1 = 0.1 Case x - 1 = x = 1.1 x = 1. The minimum height of the bridge is 1. m, and the maximum height is 1.1. THINK AND DISCUSS 1. Multiply both sides by ; split into two cases and solve each one.. An absolute-value equation can have no solutions: one solution: x + 1 = -1 x + 1 = 0 no solution -1 EXERCISES GUIDED PRACTICE 1. x = Case 1 Case x = x = - x = x = -. 9 = x + Case 1 Case 9 = x + -9 = x = x -1 = x 9 = x + 9 = x x + = - - x = Case 1 Case x = x = - x = x = - x = x = - x + = x + = () + (-) two solutions: x + 1 = -, 1 Holt McDougal Algebra 1

22 . x = 1 x = 1 x = 9 Case 1 Case x = 9 x = -9 x = 1 x = (9) 1 (9) x + Case 1 x + = x = x + = x - - = + + x - = Case 1 x - = + + x = 11 x - - = = x no solution Case x + = x = - x + + = Case x - = x = x - - = x = 0 x = 0 x = x + = -7 no solution x + 1. = x = 0. - x = 0 + x + x. = x. - x + 1. = = x = x = x = x -9 = x - = x 7 = x (-) x = x + 7 = - x + 7 = - x + 7 = - no solution 1. Let x represent the mile-marker number. x - 07 = Case 1 Case x - 07 = x - 07 = x = 09 x = 0 The walkie-takie will reach between mile marker 0 and mile marker 09. PRACTICE AND PROBLEM SOLVING 1. x = Case 1 x = Case x = - x = x = - Holt McDougal Algebra 1

23 1. x - = Case 1 Case x - = x - = x = x = -1 x = x = -1 x = 1 x = -9 x - = x - = (1) - (-9) = x = x - 1 = x - 1 Case 1 Case = x = x = x - = x 1 = x = x () 1 () x = - - x - = - - x = Case 1 Case x = x = - - x = - - x = () - -() x - 1 = x = 0 x = 0 x = 10 Case 1 Case x = 10 x = -10 x - 1 = 1 x - 1 = (10) (10) x -.0 =. Case 1 Case x -.0 =. x -.0 = x =. x = 1. x -.0 =. x -.0 = x - = Case 1 Case x - = + + x - = x = x = 0 ( ) x = ( ) ( ) x = ( x = x = 0 ) 0 x - = x - = () - (0) x + 1 = 1 Case 1 Case x + 1 = 1 x + 1 = x = 1 x = -1 x = 1 x = -1 x = x = - 1 x + 1 = 1 () x + 1 = 1 ( - 1 ) Holt McDougal Algebra 1

24 . -x + =. Case 1 -x + =. - - Case -x + = x =. -x - = -x = -.. -x - - = -. - x = -1. x =. -x + =. -x + =. -(-1.) x + 9 = x = 0 x = 0 x 0 = x = 0 x + 9 = 9 (0) = 7 - x = - x (-1)(1) = (-1)(- x ) - 1 = x no solution. x = x - = -9 no solution -(.) x + = 1 - x + = - - x = 0 x = 0 x + = = - x 0 = - x + x + x x = 0 = - x x - 1 = - - x - 1 = 0 x - 1 = x = 1 + x - 1 = Let x represent the diameter of the valve. x - = Case 1 Case x - = x - = x =.001 x =.999 The value can be between.999 mm and.001 mm. 0. n - = Case 1 Case n - = n - = n = n = - 1. x - 7 = Case 1 Case x - 7 = x - 7 = x = 9 x =. Let x represent the diameter. x -. = 0.0 Case 1 x -. = Case x -. = x =. x =. Maximum diameter is. mm; minimum diameter is. mm.. Let x represent the number of bricks. x - = 0.0() x - = 7 Case 1 Case x - = 7 x - = x = 17 x = 1 The maximum number of bricks is 17, and the minimum number of bricks is 1.. l -.1 = 0. Case 1 l -.1 = 0. Case l -.1 = l =. l =.9 No; acceptable range is from.9 in. to. in. Holt McDougal Algebra 1

25 . - (-) - = 0 Equation is x - 0 = x = 7. - (-1) - = Equation is x - = = = = =. 1 - (-1) 1-1 = 0 Equation is x - 0 = 1 x = 1. - (-) = = 1 = = - = -1 Equation is x - (-1) = x + 1 = 9. sometimes; if equation simplifies to an absolute value on one side and a positive number on the other 0. sometimes; if x is non-negative 1. always; if x is non-negative, x = x is non-negative, and if x is negative, x = -x, which is non-negative.. Let x represent the temperature of the freezer. x - =. Case 1 Case x - =. x - = x =. x = -0. The maximum temperature in the freezer is. F, and the minimum temperature in the freezer is -0. F. a. t - = b. Case 1 Case t - = t - = t = 9 t = 19 c. yes, if you change the left side so that the measured wind speed is being subtracted from t d. The measurement is correct within mi/h. a. r - b. r - = c. Case 1 Case r - = r - = r = r = 10 Least: 10 gal/min Greatest: gal/min. Possible answer: No; when an absolute-value expression equals 0, there is only one equation to solve. When an absolute-value expression equals a negative number, there is no equation to solve.. No; no matter what value of a is chosen, there will always be two solutions: a + 1 and a - 1. TEST PREP 7. C; Maximum and mimimum weights are ( + ) kg and ( - ) kg, respectively.. J; one case is n + 1 = n = B; because =. and 9 -. = 9. CHALLENGE AND EXTEND 0. P = l + w = x - + x - x - x - x = x - Case 1 - x = (x - ) - x = (x) - () - x = x x = x + x + x 10 = x 10 = x 1 = x Case -( - x) = (x - ) - - (-x) = (x) - () - + x = x x = x - x - x -9 = x -9 = x - = x The value of x is 1 in.; solutions to equation are 1 and -, but since x represents length, the negative solution is not reasonable. 1. Div. Prop. of Eq.; Subtr. Prop. of Eq.; Div. Prop. of Eq.. x = x + 1 Case 1 x 0 x = x x - x 0 = 1 no solution x = x Case x < 0 if x + 1 < 0, then -x = -(x + 1) no solution as in Case 1 if x + 1 0, then -x = x x - x -x = 1 -x - = 1 - x = - 9 Holt McDougal Algebra 1

26 SPIRAL REVIEW. =. = p = p q = q = 1. m + n = 7 - n - n m = 7 - n = 0. y + x = 1 - x - x y = 1 - x y = 1 - x y = 1 - x. c + d = e (c + d)e = ( e) e (c + d)e = (c + d)e (c + d) = (c + d) e = c + d = = y = y 7. y -. = y = S = T + R - R - R S - R = T 1. + w = x z + w (z) = x(z) z + w = xz - - w = xz -. M - N = S + N + N M = S + N - S - S M - S = N READY TO GO ON? PAGE x - = x = 1. j + = j = = m = m. 9 = g = g. Let s represent the space used on Soledad s hard drive. 1 + s = s = 7 Soledad used 7 GB of hard drive space in the first six months. h = -1 () ( h ) = ()(-1) h = = w - (-)(-.) = (-) ( w. = w -). = c = c 1 = c b =.7-0.1b -0.1 = b = Let g represent the amount of the goal. g = 00 ( ) ( g ) = ( ) (00) g = 000 The fund-raising goal was $ r + 0 = r = 10 r = 10 r = n + - n = -1 n + = n = -1 n = -1 n = k + = k = ( ) ( k ) = ( ) () k = (x - 7) = (x) + (-7) = x - = + + x = 0 x = 0 x = Let m represent the number of miles Carmen traveled m = m = m 0.0 = m = 1 Carmen traveled 1 miles. 1. x - = x + - x - x x - = + + x = x = x = 17. (x - ) = (x - ) (x) + (-) = (x) + (-) x - 1 = x - - x - x -1 = - no solution 0 Holt McDougal Algebra 1

27 1. (t - ) = (t + ) (t) + (-) = (t) + () t - = t t - t - = t = t -1 = t -9 = t 19. 7(x + ) = -7(x + ) 7(x) + 7() = -7(x) - 7() 7x + = -7x - + 7x + 7x 1x + = x = -70 1x 1 = x = - 0. Let w represent the number of weeks w = 0 + 1w + w + w 700 = 0 + 0w = 0w 0 0 = 0w 0 = w 0 + 1w = 0 + 1() = = $ Diego and Juan will both have $ in their savings accounts in weeks. 1. x + y = 1 - y - y x = 1 - y x = 1 - y 1 - y x = x = - y. j + s = t j + s + = t. r = 7 Case 1 Case r = -7 r = 7. x r = v (r) ( x r ) = (r)(v) x = rv. h + p = (k - ) h + p (k - ) = h + p = k h + p + = k. h + = 11 Case 1 Case h + = -11 h + = x + = 0 h = -1 h = 7 x + = x = - x = - x = -. 1 = 7 p = 7 p + -1 = 7 p = p + no solution -7 RATES, RATIOS, AND PROPORTIONS, PAGES 10 1 CHECK IT OUT! 1. 1 = 1 = 1 mi. 0 ft mi h = 0. ft/s 0. ft/s h 00 s..0 7 = $7.0/h Possible answe r: The cyclist travels about 0 mi in h, or 1 mi/h. There are about 000 ft in 1 mi, so this speed is equivalent to 1(000) = 7,000 ft/h. There are about 000 s in 1 h, so this is equivalent to 7,000 0 ft/s. So 0. ft/s is reasonable. 000 a. - = y y = - y = x = 1 x = 1 x = 0. ft = in. b. g + = 7 g + = 7 g = - 1 g = =.7 1 Holt McDougal Algebra 1

28 THINK AND DISCUSS 1. Possible answer: by using cross products; by multiplying both sides by to isolate t.. Possible answer: 1 in. on the map represents 1 mi. So 0. in. will represent a little more than half of 1 mi. So 11. is reasonable.. Possible answers: rate: to compare quantities with different units; unit rate: to compare prices of products of different sizes; proportion: to solve for a missing quantity; conversion factor: to convert from one set of units to another. EXERCISES GUIDED PRACTICE 1. The ratios are equivalent. x. x $ = x = $ x = $. 1 trillion = 1 x = 1 trillion 1 = trillion. x 1 = 000. x 0 1 =,9 1 x = 0 rotations/s x = 1,79 lb/cow. x 1 = /h 1 h 0 min x = 0. m/s = 0.07 page/min. 1 ft s 00 s 1 mi 1 h 0 ft.1 mi/h 1 yards 9. tablespoons 1 mi 1 tablespoon 1 gallon 170 yards = 1 mi/gal 10. z = 11. x = 1(z) = () z = 1. b = (b) = () b = 1 b = (x) = 1() x = x = 1. f + 1 = 7 (f + ) = 1(7) f + = - - f = 7 f = = d -1(d) = () -d = 1 d = = 7 x -(x) = 7(9) -x = x = h = 1 (h) = 10(1) h = 10 h = = s - 1 1(s - ) = 1() 1s - = + + 1s = 91 s =. 17. s - = 7 1(s - ) = (7) s - = s = 19. Let h be the height of the Altar Stone. = h.9 (h) =.9() h = 1.7 h =.9 m PRACTICE AND PROBLEM SOLVING 0. x = 1. x 9 1,000 = 00 1(x) = (9) x = in.. lb =. lb/gal gal. $0.0 = $0.90/oz 1 oz ft days = 11.9 ft days 1 day 1 in. h 1 ft 1.9 in./h 00(x) = 1,000() 00x =,000 x = ft in days is roughly equivalent to 1 ft in 7 h, or 1 7 = ft/h. There are 1 in. in 1 ft, so this speed is equivalent to 1 = in./h. So 1.9 in./h is reasonable.. 9 m/s = 9 m 00 s 1 km 1 s 1 h 0 m = 9. km/h. v = 7. = y (v) = (1) v = (y) = () y = 0 v = y = 10 Holt McDougal Algebra 1

29 . h = - (-)(h) = () (-)h (-) = 1 (-) h = t 9 = (t) = 1(9) 10t 10 = 9 10 t = 0.9 x. = (x) = (7.) 0x 0 = 0 0 x =. a = (a) = (17) 1a 1 = 10 1 a =.. = x (x) = () x = 00 x = 0. 1 = 0 mm x x = 0 mm x = 0. mm So, the dust mite is 0. mm long. x = b (b + 7) = 0() 10b + 70 = b = b 10 = b = = q - (q - ) = () q - = q = q = q = 1. k = 1 (k) = (1) k = k = = x + 1 9(x + 1) = () 9x + 9 = x = 1 9x 9 = 9 x = = n (n - ) = 0() 19n - 9 = n = 19n 19 = 19 n = = 0 0 0(x) = 0(70) 0x 0 = 00 0 x = $ 70 euro will be worth more than $0, so $ is reasonable = c 0(c) = () 0c 0 = 00 0 c = carp 1. Possible answer: The denominator in the first ratio is the number of varsity members, but in the second ratio it is the total number of people on the team. a. v = d t m 10. s 9. m/s 00 m 9.9 m/s 1. s 00 m 11. s 7.0 m/s 000 m.7 m/s.7 s b. The m race has the fastest speed, whereas, the 000 m race has the slowest speed. c. Possible answer: Runners can maintain a very high speed for a short distance, but over a longer distance their speed drops.. 1 frames - 0 frames hours = 1. frames/h. x - 1 = x + 1. m = m + 7 (x - 1) = (x + 1) 7(m) = (m + ) x - = x + - x + - x + 7m = m m - m x = x = m = 1 m = x = m =. x - = 7. a x - = a - 0 1(x - ) = (x - ) x - = x x + - x + 0(a) = (a - ) 0a = a - - a - a -x = - -x - = a = - - a - = x = a = - 7. y = 1 9. n + = n - 1 y + (y + ) = 1(y) (n+) = (n - 1) y + = y n + = n - - y - y - n - - n - = 9y -n = -11 -n - = 0. 9 = 9y 9 y = 9 y = y - 1 1(y - 1) = 1(y) y - 1 = y - y y + 1 y = 1 y = y = - n = 1 1. n = n + (n + ) = (n) n + = n - n - - n - -n = - -n - = - - n = Holt McDougal Algebra 1

30 . t - - = t +. d + = d + 1 (t - ) = -(t + ) (d + 1) = (d + ) 10t - = -t - d + = d t + + t + - d - - d - 1t = 0 -d = - t = 0 -d -1 = - -1 d = x +. = x 1 (x + ) = 1(x) 9x + 1 = 1x - 1x - 1-1x - 1 -x = -1 -x - = -1 - x =. n = n - (n - ) = (n) 1n - 10 = 1n - 1n n n = 10 -n -1 = 10-1 n = APPLICATIONS OF PROPORTIONS, PAGES 17 1 CHECK IT OUT! 1. 7 = x (x) = 7() x = 1 x =. in. b.. x =..(x) =.().x. = 1. x = ft a. 10 x = 19 (x) = 10(19) x = 9,0 x = 0 cm. The ratio of the perimeters is equal to the ratio of the corresponding sides. THINK AND DISCUSS 1. Possible answer: a model airplane and the real airplane it represents; a baseball and a softball; a photograph and its enlargement. EXERCISES GUIDED PRACTICE 1. They are the same shape but not necessarily the same size.. = 7. x = x x = x = x = 0 x = 0 x =. m x = 10 ft. h =. 1.h = 70.h. = 70. h = 0 ft. The ratio of the areas is the square of the ratio of the corresponding side lengths. PRACTICE AND PROBLEM SOLVING. = 7. x 10 = x x = 1 x = 10x = x 10 = x = m x = 7 in.. 1 h = 0 0h = 7 0h 0 = 7 0 h = 7. ft 9. The ratio of the perimeters is equal to the ratio of the corresponding side lengths. 10. = x x = 1 x = 1 x = ft Possible answer: The width is less than the length for the baby blanket, so the width on the mother s blanket will also be less than the length. Since ft is less than ft, ft is a reasonable answer ) x = ( 10 x = x = 0 ft = ( 1) x = x x = 1. 1) 1 = ( x 1 = x 1 x = x = x = 1. 1 x = 7 x = 7 x = 9 m Holt McDougal Algebra 1

31 1 1. x = 10 10x = 10x 10 = 10 x =. ft 17. x = + x = x = x = cm 1. 7 x = 10 x = 70 x = 70 x =.7 ft 1. x = 0 x = 100 x = 100 x = 00 ft 19. Possible answer: No; a 1-in. pizza actually has times the area, so the cost should be times as much. 0a. ; 1 ; ; 7 ; 9 b. 0.1; 0.1; 0.; 0.7; 0.9 c. The decimal is the percent with its decimal point moved two places to the left and the % symbol removed x =.. 0 x = 0 7.x =.x. = 0x = 0 0x. 0 = 0 0 x = 1 m x = 9 m. k TEST PREP. C; The radius of the first ball is twice the radius of the second. Since =, to find the volume of the smaller sphere, divide 00 by.. G; Students can look at the first ratio in each similarity statement and check that the letters S and G are in the same position as M and W. Only choice G has them placed incorrectly.. x = 7. x = 1.7 x = 1.7 x =. cm CHALLENGE AND EXTEND 7. w =..w = 1.w. = 1. w = m y = y = y = y = m 10 x = x = 0 x = 0 x = 7. m. b 9. Let A represent the area of rectangle B, w represent width, and P represent perimeter. A 0.19 = ( ) A 0.19 = 9 9A = A 9 = A = 1. cm A = l w w = A.1 w = 1..1 w =. cm P = (l + w) P = (.1 +.) P = (.) P = 1. cm SPIRAL REVIEW = = = = -. (-, -); (-1, -); (0, 0); (1, ); (, ). (-, ); (-1, 1); (0, 0); (1, 1); (, ). (-, ); (-1, 7); (0, ); (1, ); (, ) 7. (-, -7); (-1, -); (0, -1); (1, ); (, ). x = 9. x = x = 1 x = 9 x = x = Holt McDougal Algebra 1

32 0. 1 = - f f = - f = = x x + 10 = x = 10x 10 = 10 x =. -9 PERCENTS, PAGES 1 1 CHECK IT OUT! 1a. x 0 = 0 x = x = 1 c. x = x = 1 x = 1. b. 7 = x 9 7 = 9x 7 9 = 9x 9 = x 00% = x b. = 1% of x = 0.1 x 0.1 = 0.1x 0.1 0% = x b. x = 10% of x =.1() x = 1. a. 7 = x x = 700 x = 700 x = 0% a. 90 x = 10 10x = x 10 = x = 7 x. = x = x = x 10.0 The bracelet has approximately 10 karats. THINK AND DISCUSS 1. If the part is greater than the whole, then the percent is greater than %. Possible answer: If the whole is 0 and the part is, then the percent is 1%.. Greater than ; % is less than %, so is less than the whole. So the whole is greater than.. EXERCISES GUIDED PRACTICE 1. Possible answer: a number compared to.. x 0 = 7. x = 1.% of 1 x = 000 x = 0.1(1) x = 0 x = 1. x = 11% of 7 x = 1.1(7) x =.. 0 = x 0x = 00 0x 0 = 00 0 x =.%. 7 0 = x 0x = 700 0x 0 = x = 190% 10. x = x = 00 x = 00 x = x = 10 10x = x 10 = x = x = x = 00 x = g PRACTICE AND PROBLEM SOLVING 1. 0% of 0 = 0.(0) = 17. % of 00 = 0.00(00) = = x x = x = x = % = x x = 1 x = 1 x = 0%. x = 70% of x = 0.7() x = = x x = 1,000 x = 1,000 x = 0% 9. = x x = x = x = 1.% x = x = 700 x = 700 x = = 10 x 10x = 70 10x 10 = x = 1. % of 90 = 0.(90) = % of 0 =.1(0) = = x 9x = 700 9x 9 = x = 00%. 7 = x 7x = 00 7x 7 = 00 7 x 7.% Holt McDougal Algebra 1

33 . x = 90 90x = 00 90x 90 = x = 0. = x.x = 00.x. = 00. x x 00 = x = 10,000 x = 10,000 x = mg.. x = x = 0 x = 0 x = = 9 x 1x = 900 1x 1 = x = 7.. = 1. = 1% = % = 7% 1. = 0.0 = %. 7 7 = 1 = %. 0. = %. 1 = 0.1 = 1%. 0. = % = 0.% = 0. = %. % = 0.; % = 0.; % = 90.0; % = 1.1;. % = 0.0; 0 7. % = 0.00; % = 0.1;. 7.% = 0.7; % = 0.9; 9. 1.% = 0.01; 00. Estimates: 0; 00; Actual: 9.; is greater than % because % = ,,.%, 0, , 1%,, 11%, , %,. 0., 9%,. 0.9, 9,, %, = x x = 10 x = 10 x 9% a. 0 ( ) = 0 00 = 0.0 = 0% x b. ( ) = x = 0,000 x = 00; action c. ( ) = 00 = 0.0 = % d. 0 - ( ) = 9 00 = 0..9% 1,0,. (1,0, + 1,,) = 1,0, 1,1, % are male 1,, (1,0, + 1,,) = 1,, 1,1, % are female 7. 1% of 00 % of 1 is. is 1. % of 00 is. % of is. % of 0 is. 0% of is 1. % of is 1. 1.% of 9 is 1. % of 0 is 0. 0% of 0 is 0. % of 0 is 0. 00% of 10 is 0. Possible answer: As the percent doubles, the missing value is divided by two, and vice versa.. Possible answer: The equation method may require fewer steps. The proportion method is good if you cannot remember how to set up the equation, because the proportion method will work for any type of percent problem. 9a. x 90 = 0 x = 00 x = 00 x = $ b. $90 - $ = $ Possible answer: The discount is 0%, so the sale price will be more than half of $90, or $. So $ is reasonable. c. Possible answer: $90 is divided into 10 groups, and each group represents 10%. So of the groups represent 0%. 7 Holt McDougal Algebra 1