Chapter Maintaining Mathematical Proficiency (p. ). + ( ) = 7. 0 + ( ) =. 6 + = 8. 9 ( ) = 9 + =. 6 = + ( 6) = 7 6. ( 7) = + 7 = 7. 7 + = 8. 8 + ( ) = 9. = + ( ) = 0. (8) =. 7 ( 9) = 6. ( 7) = 8. ( 6) =. 6 = 8. ( ) = 6. 6 8 = 8 7. 6 6 = 6 8. ( ) = 9. a. Sample answer: To add integers with the same sign, add their absolute values, and the sum has the same sign as both addends. To add integers with different signs, subtract their absolute values, and the difference has the same sign as the addend with the greatest absolute value; 6 + = b. Sample answer: To subtract integers, change the subtraction sign to an addition sign, and change the integer following the sign to its opposite. Then, follow the rules for adding integers; ( ) = + = 8 c. Sample answer: To multiply integers, multiply their absolute values. If the integers have the same sign, then the product is positive. If the integers have different signs, then the product is negative; ( 6)( ) = d. Sample answer: To divide integers, divide their absolute values. If the integers have the same sign, then the quotient is positive. If the integers have different signs, then the quotient is negative; = Chapter Mathematical Practices (p. ). Population change = 0 million 80 million = 0 million people Time change = 00 000 = 0 years 0 million people Rate of change = 0 years 0 mi. Gas mileage = 8 gal = million people per year = 0 mi/gal. 8 in. =. ft Volume = wh = ( ft) ( ft) (. ft) =. ft Amount of water = (. ft ) = 6.87 ft 6.87 ft Drain time = ft /min = 6.87 min It takes about 7 minutes for the water to drain.. Explorations (p. ). Quadrilateral m A (degrees) m B (degrees) m C (degrees) m D (degrees) m A + m B + m C + m D a. 0 90 9 68 60 b. 6 7 8 90 60 c. 9 79 7 60 Sample answer: Because a protractor is used, the measurements are precise.. Conjecture: The sum of the angle measures of a quadrilateral is 60. Sample answer: D A C 00 + 0 + 0 + 80 = 60 A D 90 + 60 + + 9 = 60 D C A C 0 + 0 + 7 + 0 = 60 Divide the quadrilateral into two triangles. The sum of the angle measures of a triangle is 80, so the sum of the angle measures of a quadrilateral is (80 ) = 60. B B B Copyright Big Ideas Learning, LLC Algebra
. a. x + 80 + 8 + 00 = 60 So, x = 9. x + 6 = 60 6 6 x = 9 b. x + 78 + 60 + 7 = 60 So, x = 0. x + 0 = 60 0 0 x = 0 c. x + 90 + 0 + 90 = 60 So, x = 0. x + 0 = 60 0 0 x = 0. Sample answer: If you notice a pattern, you can use inductive reasoning to write a rule. Then you can test your rule using several examples. You can use the rule to write an equation that can be used to solve a problem.. Sample answer: The corners can be arranged so the angles complete a full circle, which is 60.. 6. = p +.9 Write the equation...9.9 Subtract.9 from each side. 0. = p Simplify. Check: 6. = p +.9 6. =? 0. +.9 6. = 6. The solution is p = 0.. ( y y = 6 Write the equation. ( 6) Multiply each side by. ) = y = 8 Simplify. y Check: = 6 8 =? 6 6 = 6 The solution is y = 8.. 9π = πx Write the equation. 9π π = πx π Divide each side by π. 9 = x Simplify. Check: 9π = πx 9π =? π(9) 9π = 9π The solution is x = 9. 60. Monitoring Progress (pp. 7). n + = 7 Write the equation. Subtract from each side. n = 0 Check: n + = 7 0 + =? 7 Simplify. 7 = 7 The solution is n = 0.. g = + + g = Write the equation. Add to each side. Simplify. Check: g = =? = The solution is g =. 6. 0.0 w =. Write the equation. 0.0 w 0.0 =. 0.0 w = 8 Check: 0.0 w =. 0.0(8) =?. Divide each side by 0.0. Simplify.. =. The solution is w = 8. 7. Let t be the time it would take to run 00 meters. Use the Distance Formula. d = r t 00 = 0. t 00 0. = 0.t 0. 8.6 t Usain Bolt would run 00 meters in about 8.6 seconds. Algebra Copyright Big Ideas Learning, LLC
8. Words: Your recorded balance Variable: Forgotten check = Statement balance Let c be the amount of the forgotten check. Equation: 68 c = 6 68 c = 6 68 68 c = c = The check you forgot to record was for $.. Exercises (pp. 8 0) Vocabulary and Core Concept Check. Addition, +, and subtraction,, are inverses of each other. Multiplication,, and division,, are inverses of each other.. x = 0 x = x = 0 x = x = x = yes; The equations are equivalent because they have the same solution, x =.. Division Property of Equality; In order to write an equivalent equation that has x by itself on one side, you must undo multiplying by. So, you would divide each side by.. The equation x 6 = does not belong. Sample answer: You would use the Addition Property of Equality to solve it, whereas you would use the Multiplication Property of Equality to solve the other three equations. Monitoring Progress and Modeling with Mathematics. x + = 8 Write the equation. Subtract from each side. x = Simplify. Check: x + = 8 + =? 8 8 = 8 The solution is x =. 6. m + 9 = Write the equation. 9 9 Subtract 9 from each side. m = 7 Simplify. Check: m + 9 = 7 + 9 =? = The solution is m = 7. 7. y = Write the equation. + + Add to each side. y = 7 Simplify. Check: y = 7 =? = The solution is y = 7. 8. s = Write the equation. + + Add to each side. s = Simplify. Check: s = =? = The solution is s =. 9. w + = Write the equation. Subtract from each side. w = 7 Simplify. Check: w + = 7 + =? = The solution is w = 7. 0. n 6 = 7 Write the equation. + 6 + 6 Add 6 to each side. n = Simplify. Check: n 6 = 7 6 =? 7 7 = 7 The solution is n =.. = p Write the equation. + + Add to each side. = p Simplify. Check: = p =? = The solution is p =.. 0 = + q Write the equation. Subtract from each side. = q Simplify. Check: 0 = + q 0 =? + ( ) 0 = 0 The solution is q =. Copyright Big Ideas Learning, LLC Algebra
. r + ( 8) = 0 Write the equation. ( 8) ( 8) Subtract 8 from each side. r = 0 + 8 r = 8 Check: r + ( 8) = 0 8 + ( 8) =? 0 0 = 0 The solution is r = 8. Rewrite subtraction. Simplify.. t ( ) = 9 Write the equation. t + = 9 Rewrite subtraction. t = Simplify. Check: t ( ) = 9 ( ) =? 9 + =? 9 Subtract from each side. 9 = 9 The solution is t =.. Words: Discounted ticket price = Original price.9 Variable: Let p be the original price. Equation: = p.9 = p.9 +.9 +.9 6.9 = p The original price of an amusement park ticket is $6.9. 6. Words: Your final score Variable: + = Friend s final score Let x be your final score. Equation: x + = 9 x + = 9 Your final score is 8 points. 7. x + 00 + 0 + 00 = 60 So, x = 0. x + 0 = 60 0 0 x = 0 8. x + 8 + 77 + 0 = 60 So, x = 8. x + 7 = 60 7 7 x = 8 x = 8 9. x + + 9 + 76 = 60 So, x = 70. x + 90 = 60 90 90 x = 70 0. x + 60 + + 8 = 60 So, x = 00. x + 60 = 60 60 60 x = 00. g = 0 Write the equation. g = 0 g = Check: g = 0 () =? 0 Divide each side by. Simplify. 0 = 0 The solution is g =.. q = Write the equation. q = Divide each side by. q = Simplify. Check: q = () =? = The solution is q =.. p = Write the equation. (p ) = () Multiply each side by. p = Check: p = =? = The solution is p =. Simplify.. y 7 = Write the equation. 7 ( y 7) = 7 () Multiply each side by 7. y = 7 Check: y 7 = 7 7 =? = The solution is y = 7. Simplify. Algebra Copyright Big Ideas Learning, LLC
. 8r = 6 Write the equation. 8r 8 = 6 8 Divide each side by 8. r = 8 Simplify. Check: 8r = 6 8( 8) =? 6 6 = 6 The solution is r = 8. 6. x ( ) = 8 Write the equation. 7. 8. [x ( )] = 8 Multiply each side by. x = 6 Check: x ( ) = 8 6 ( ) =? 8 8 = 8 The solution is x = 6. 6 ( x Simplify. x = 8 Write the equation. 6 6) = 6 x = 8 Check: x 6 = 8 8 6 =? 8 8 = 8 The solution is x = 8. ( w w = 6 ) = w = 8 Check: w = 6 8 =? 6 (8) Multiply each side by 6. 6 = 6 Simplify. The solution is w = 8. Write the equation. (6) Multiply each side by. Simplify. 9. = 9s Write the equation. 9 = 9s 9 Divide each side by 9. 6 = s Simplify. Check: = 9s =? 9( 6) = The solution is s = 6. 0. 7 = t 7 7) 7 ( 7) = 7 ( t 9 = t Check: 7 = t 7 7 =? 9 7 7 = 7 The solution is t = 9.. + t = t =, or Check: + t = + ( = )? = The solution is t =.. b 6 = 6. + 6 + 6 b = 8 6, or Check: b 6 = 6 8 6 6 =? 6 6 = 6 The solution is b =. 7 m = 6 7 7 m = 7 6 m = Check: 7 m = 6 7 () =? 6 6 = 6 The solution is m =. Write the equation. Multiply each side by 7. Simplify. Copyright Big Ideas Learning, LLC Algebra
. y = ( y ) = y = 0 Check: y = ( 0) =? = The solution is y = 0... = a 0. + 0. + 0..6 = a Check:. = a 0.. =?.6 0.. =. The solution is a =.6. 6. f + π = 7π π π f = π Check: f + π = 7π π + π =? 7π 7π = 7π The solution is f = π. 7. 08π = 6π j 08π 6π 8 = j = 6π j 6π Check: 08π = 6π j 08π =? 6π ( 8) 08π = 08π The solution is j = 8. 8. x ( ) =. [x ( )] =. x =.8 Check: x ( ) =..8 ( ) =?.. =. The solution is x =.8. 9. A positive 0.8 should have been added to each side. 0.8 + r =.6 + 0.8 + 0.8 r =. The solution is r =.. 0. Each side should have been multiplied by. m = ( m ) = ( ) m = The solution is m =.. C; Because each carton contains 8 eggs, the total number of eggs is equal to the product of 8 and the number of cartons, x. 8x = 6 8x 8 = 6 8 x = 9 The baker orders 9 cartons of eggs.. Words: Temperature at p.m. Change in temperature = Temperature at 0 p.m. Variable: Let T be the change in temperature. Equation: 0 T = 0 T = 0 0 T = T = The temperature fell F from p.m. to 0 p.m.. Words: Length =.9 Width Variable: Let w be the width. Equation: 9. =.9 w 9. =.9w 9. =.9w.9.9 = w The American flag is feet wide. 6 Algebra Copyright Big Ideas Learning, LLC
. Words: Current balance Variable: = $08 + Balance years ago Let b be the balance years ago. Equation: 708 = 08 + b 708 = 08 + b 08 08 00 = b The balance years ago was $00.. Multiplication Property of Equality [ x = x ] + x = x + 6. a. Words: Total area = Area of rectangle Variable: Let A be the area of one rectangle. + Area of square Equation: 8 = A + 0.A 8 = A + 0.A 8 =.A 8. =.A. 8 = A Each rectangular mat is 8 square feet. b. Guess: length = 8 ft, width = ft Check: A = w 8 =? 8() 8 Revise: length = 6 ft, width = ft Check: A = w 8 =? 6() 8 = 8 Each rectangular mat is 6 feet by feet. 7. a. Words: Total spent Variable: = Number of CDs Amount you spend per CD Let p be the amount you spend on each CD. Equation: 0.0 = p 0.0 = p 0.0 = p 7.6 = p You spend $7.60 on each CD. b. Words: Amount you spend per CD = 80% Original price Variable: Let p be the original price. Equation: 7.6 = 0.80 p 7.6 = 0.8 p 7.6 0.8 = 0.8 p 0.8 9. = p Each CD costs $9.0. Because CDs at the original price cost (9.) = $8.0, $ is not enough to buy them all. 8. Equation Value of x Reason x c = 0 increases Because x = c, as c increases, so does x. cx = decreases Because x =, as c increases, c x decreases. cx = c stays the same Because x =, as c increases, x stays the same. x c = increases Because x = c, as c increases, so does x. x c = 0 cx = cx = c +c +c cx c = c x = c x = c cx c = c c x = x c = c x c = c x = c Copyright Big Ideas Learning, LLC Algebra 7
9. a. x = 0 x = x = x = When a = and b = 0, x is a positive integer. b. x = 9 x = x = x = When a = and b = 9, x is a negative integer. 0. a. The entire circle represents 00%. b. The percent of the partitions of a circle graph should sum to 00 percent. You can solve the equation to find x, the percent of cats. 7 + 9 + + 8 + x = 00 69 + x = 00 69 69 x = At a local pet store, % of the animals sold are cats.. Let g be the number of girls and b be the number of boys in the marching band. 6 g = 6 7 b = 0 6 6 g = 6 6 7 7 b = 7 0 g = 6 b = 6 girls + boys = 7 students The marching band has 7 students.. Sample answer: A game has 0 pieces, each player should get the same number of pieces, and all 0 pieces should be used. If you and of your friends are playing, how many pieces should each player get? Let x be how many pieces each player should get. x = 0 x = 0 x = 6 You and your friends should each get 6 pieces.. V = Bh 8π = B(7) 8π 7 = 7B 7 π = B The area of the base of the cylinder is π square inches.. V = Bh = 7h 7 = 7h 7 9 = h The height of the rectangular prism is 9 centimeters.. V = Bh π = B() (π) = ( ) B 9π = B The area of the base of the cone is 9π square meters. 6. V = Bh 7. a. = 0 h = 0h 0 = 0h 0. = h The height of the square pyramid is. feet. Batting average = Number of hits Number of at-bats h.96 = 6 h 6 h 6 (.96) = 6 Player A had hits in the 0 regular season. b. no; If player B had fewer hits, then he must have had fewer at-bats in order to have a greater batting average. Maintaining Mathematical Proficiency 8. 8(y + ) = 8 y + 8 = 8y + 9. x + 6( ) + = 6 x + 6 + 6 = 6 x + + 0 6 = 6 x + + 0 = 6 x + = 6 x + 8 Algebra Copyright Big Ideas Learning, LLC
60. (m + + n) = m + + n = m + + n = m + n + 6. (p + q + 6) = p + q + 6 6. L 60 min min h 6. = 8p + 6q + = 00 L h 68 mi h h 60 min min 60 sec = 68 mi 600 sec 6. 7 gal min min 60 sec qt gal 8 qt = 60 sec 0.7qt sec 6. 8 km 60 min mi 80 mi min h =.6 km.6 h. Explorations (p.) 0.0 mi sec 98. mi h. a. (0 + x) + 9x + 0 = 80 Write the equation. 0 + x + 9x + 0 = 80 Associative Property of Addition x + 9x + 0 + 0 = 80 0x + 60 = 80 60 60 0x = 0 0x 0 = 0 0 x = Commutative Property of Addition Combine like terms. Subtract 60 from each side. Simplify. Divide each side by 0. Simplify. So, x = and the measures of the angles of the triangle are 0, 9x = (9 ) = 08, and (0 + x) = (0 + ) =. b. (x + 0) + (x + 0) + 0 = 80 Write the equation. x + 0 + x + 0 + 0 = 80 Associative Property of Addition x + x + 0 + 0 + 0 = 80 Commutative Property of Addition x + 80 = 80 Combine like terms. 80 80 Subtract 80 from each side. x = 00 Simplify. x = 00 x = 0 Divide each side by. Simplify, So, x = 0 and the measures of the angles of the triangle are 0, (x + 0) = (0 + 0) = 70, and (x + 0) = (0 + 0) = 60. c. 0 + (x + 0) + (x + 0) + x = 60 Write the equation. d. 0 + x + 0 + x + 0 + x = 60 Associative Property of Addition x + x + x + 0 + 0 + 0 = 60 x + 00 = 60 Commutative Property of Addition Combine like terms. 00 00 Subtract 00 from each side. x = 60 x = 60 x = Simplify. Divide each side by. Simplify. So, x = and the measures of the angles of the quadrilateral are 0, (x + 0) = ( + 0) =, (x + 0) = ( + 0) =, and x =. (x 7) + (x + ) + (x + ) + x = 60 Write the equation. x 7 + x + + x + + x = 60 x + x + x + x 7 + + = 60 x + 60 = 60 Associative Property of Addition Commutative Property of Addition Combine like terms. 60 60 Subtract 60 from each side. x = 00 x = 00 x = 7 So, x = 7 and the measures of the angles of the quadrilateral are (x 7) = (7 7) = 8, (x + ) = (7 + ) = 0, (x + ) = (7 + ) = 7, and x = 7. Simplify. Divide each side by. Simplify. Copyright Big Ideas Learning, LLC Algebra 9
e. (x + ) + (x + ) + (8x + 8) Write the + (x + 0) + (x + ) = 0 equation. f. x + + x + + 8x Associative + 8 + x + 0 + x + = 0 Property of Addition x + x + 8x + x + x Commutative + + + 8 + 0 + = 0 Property of Addition x + 0 = 0 Combine like terms. 0 0 Subtract 0 from each side. x = 00 Simplify. x = 00 Divide each side by. x = 0 Simplify. So, x = 0 and the measures of the angles of the pentagon are (x + ) = ( 0 + ) = 0, (x + ) = ( 0 + ) = 6, (8x + 8) = (8 0 + 8) = 68, (x + 0) = ( 0 + 0) = 0, and (x + ) = ( 0 + ) = 9. (x + 6) + (x + 8) + (x 8) Write the + (x 7) + (x + ) = 70 equation. 6x + + (x + 8) + (x 8) Distributive + (x 7) + (x + ) = 70 Property 6x + + x + 8 + x 8 Associative + x 7 + x + = 70 Property of Addition 6x + x + x + x + x + Commutative + 8 8 7 + = 70 Property of Addition 7x + 0 = 70 Combine like terms. 0 0 Subtract 0 from each side. 7x = 680 7x 7 = 680 7 x = 0 Simplify. Divide each side by 7. Simplify. So, x = 0 and the measures of the angles of the hexagon are (x + 6) = ( 0 + 6) = 6, 6, (x + 8) = ( 0 + 8) = 88, (x 8) = ( 0 8) =, (x 7) = ( 0 7) =, and (x + ) = ( 0 + ) = 0. Sample answer: You can check the measures of the angles using a protractor. You can also check to make sure the angle measures add up to the sum given by the formula.. a. Answer should include, but is not limited to: Check that polygons have or more straight sides of varying lengths and that all are closed figures. Suggest that students draw large polygons, because it will be easier to measure the angles. b. Answer should include, but is not limited to: The sum of the angle measures of each polygon should satisfy the formula S = 80(n ). Some might be a little off due to rounding. Have students round to the nearest whole number of degrees. c. Answer should include, but is not limited to: The same value for x should satisfy each expression for the angle measures of the polygon. d. and e. Answer should include, but is not limited to: Partners should confirm that the calculated measures of the angles are the same as (or at least close to) the measures obtained with a protractor. In addition, the sum of the calculated measures of the polygon should satisfy the formula S = 80(n ).. Sample answer: If you notice a pattern, you can use inductive reasoning to write a rule. Then you can test your rule using several examples. You can use the rule to write an equation that can be used to solve a problem.. Connecting a vertex with each of the other vertices in a polygon creates n triangles, each of which has a total angle measure of 80.. S = 80(n ) 080 = 80(n ) 080 80(n ) = 80 80 6 = n + + 8 = n 8 sides; Use the formula S = 80(n ) and inverse operations to work backward and find that a polygon, whose angle measures sum to 080, is an octagon.. Monitoring Progress (pp. ). n + = 9 n = 6 n = 6 n = Check: n + = 9 ( ) + =? 9 6 + =? 9 9 = 9 The solution is n =. 0 Algebra Copyright Big Ideas Learning, LLC
. = c + + 0 = c ( 0) = c 0 = c Check: = c =? ( 0) =? 0 = The solution is c = 0.. x 0x + = 8 x + = 8 x = 6 x = 6 x = Check: x 0x + = 8 ( 0 ) ( + = )? 8 + + =? 8 The solution is x =.. (x + ) + 6 = 9 (x) + () + 6 = 9 x + + 6 = 9 x + 9 = 9 9 9 x = 8 x = 8 x = 6 Check: (x + ) + 6 = 9 [( 6) + ] + 6 =? 9 ( ) + 6 =? 9 + 6 =? 9 The solution is x = 6. 8 = 8 9 = 9. = + (d ) = + (d) () = + 8d = 8d 7 + 7 + 7 = 8d 8 = 8d 8.7 = d Check: = + (d ) =? + [(.7) ] =? + (. ) =? + (.) =? + 0 = The solution is d =.7. 6. = (y ) + y = (y) ( ) + y = y + 8 + y = y + 8 8 8 = y Check: = (y ) + y =? ( ) + () =? () + =? + = The solution is y =. 7. x( ) x = x() x = x x = x = Check: x( ) x = ()( ) () =? ()() =? 0 =? The solution is x =. = Copyright Big Ideas Learning, LLC Algebra
8. (m + ) m = (m) + ( )() m = 8m + ( 0) m = m 0 = + 0 + 0 m = m = m = Check: (m + ) m = [( ) + ] ( ) =? ( 0 + ) + =? ( ) + =? The solution is m =. 0 + =? = 9. ( x) + ( x) = () (x) + () (x) = x + 6 x = 7x + = 7x = 7 7x 7 = 7 7 x = Check: ( x) + ( x) = ( ) + ( ) =? The solution is x =. 0. d = n + 6 0 = n + 6 6 6 = n = n 8 = n () + () =? 0 + =? = A fire hose needs 8 pounds per square inch of water pressure to reach a fire 0 feet away.. Words: Perimeter = Width + Three times the width Variable: Let w be the width of the rectangular pen. Equation: 96 = w + (w) 96 = w + (w) 96 = w + 6w 96 = 8w 96 8 = 8w 8 = w = w = () = 6 The pen should be 6 feet by feet.. Exercises (pp. 6 8) Vocabulary and Core Concept Check. To solve the equation x + x = 0, first combine x and x because they are like terms.. Sample answer: One way to solve the equation (x ) = 0 is to first use the Distributive Property to eliminate the parentheses. Then undo subtraction to isolate the x-term. Finally, undo the multiplication to solve for x. Another method is to first undo multiplication and use the Division Property of Equality to divide each side by. Then undo subtraction to isolate the x-term. Finally, undo multiplication again to solve for x. Monitoring Progress and Modeling with Mathematics. w + 7 = 9 7 7 w = w = w = Check: w + 7 = 9 () + 7 =? 9 + 7 =? 9 9 = 9 The solution is w =. Algebra Copyright Big Ideas Learning, LLC
. g = + + g = 6 g = 6 g = 8 Check: g = (8) =? 6 =? = The solution is g = 8.. = q = q = q = q Check: = q =? = The solution is q =. 6. 0 = 7 m 7 7 = m = m = m Check: 0 = 7 m 0 =? 7 ( ) 0 =? 7 + 0 = 0 The solution is m =. 7. = z + + z 8 = ) 8 = ( z = z z Check: = =? =? 8 = The solution is z =. 8. a + = 6 a = a = a = 6 Check: a + = 6 6 + =? 6 + =? 6 6 = 6 The solution is a = 6. h + 6 9. = h + 6 = h + 6 = 0 6 6 h = Check: h + 6 = + 6 =? 0 =? = The solution is h =. Copyright Big Ideas Learning, LLC Algebra
0. d 8 = d 8 = d 8 = + 8 + 8 d = 6 Check: d 8 = 6 8 =? =? = The solution is d = 6.. 8y + y = y = y = y = Check: 8y + y = 8() + () =? + =? = The solution is y =.. 6 = n n 6 = 9n 6 9 = 9n 9 = n Check: 6 = n n 6 =? () () 6 =? 6 6 = 6 The solution is n =.. v + 0v + = 80 v + = 80 v = 66 v = 66 v = Check: v + 0v + = 80 () + 0() + =? 80 6 + 0 + =? 80 The solution is v =.. 6c 8 c = 6 c 8 = 6 + 8 + 8 c = 8 c = 8 c = 80 = 80 Check: 6c 8 c = 6 6( ) 8 ( ) =? 6 8 + =? 6 0 + =? 6 6 = 6 The solution is c =.. a = 00t + 600,000 = 00t + 600 600 600 0,00 = 00t 0,00 00 = 00t 00 6 = t The plane is at an altitude of,000 feet 6 minutes after liftoff. Algebra Copyright Big Ideas Learning, LLC
6. Words: Repair bill = Parts cost + Labor cost per hour Variable: Let t be the number of hours of labor spent repairing the car. Hours of labor Equation: = 6 + 8 t = 6 + 8t 6 6 88 = 8t 88 8 = 8t 8 6 = t The repair bill includes charges for 6 hours of labor. 7. (z + ) = (z) + () = z + 0 = 0 0 z = z = z = Check: (z + ) = ( + ) =? (8) =? = The solution is z =. 8. (g ) = 0 (g) ( ) = 0 8g + 6 = 0 6 6 8g = 8g 8 = 8 g = Check: (g ) = 0 [( ) ] =? 0 ( ) =? 0 ( ) =? 0 0 = 0 The solution is g =. 9. 6 + (m + ) = 6 6 + (m) + () = 6 6 + m + = 6 m + = 6 m = m = m = Check: 6 + (m +) = 6 6 + ( + ) =? 6 6 + () =? 6 6 + 0 =? 6 The solution is m =. 6 = 6 0. h + ( h) = h + () (h) = h + h = h + = h = 7 h = 7 h = 9 Check: h + ( h) = ( 9) + [ ( 9)] =? + ( + 9) =? + (0) =? + 0 =? The solution is h = 9. = Copyright Big Ideas Learning, LLC Algebra
. 7 = c (6 c) 7 = c (6) ( c) 7 = c 8 + 6c 7 = 9c 8 + 8 + 8 = 9c 9 = 9c 9 = c Check: 7 = c (6 c) 7 =? () [6 ()] 7 =? (6 0) 7 =? ( ) 7 =? + 7 = 7 The solution is c =.. = y (y 7) = y (y) ( 7) = y 0y + = y + 8 = y 8 = y 9 = y Check: = y (y 7) =? ( 9) [( 9) 7] =? 8 ( 8 7) =? 8 ( ) =? 8 + = The solution is y = 9.. ( + x) + (x 6) = () + ( )(x) + (x) (6) = 9 x + x = x = + + x = 9 Check: ( + x) + (x 6) = ( + 9) + (9 6) =? The solution is x = 9. () + () =? 96 + 9 =? =. (r + 9) ( r) = (r) + (9) () ( r) = r + + r = 7r + = 7r = 7r 7 = 7 Check: (r + 9) ( r) = ( 6 + 9) [ ( 6)] =? () ( + 6) =? The solution is r = 6.. + k + k = 80 k + = 80 k = k = k = (7) =? =? r = 6 = So, k = and the measures of the angles of the triangle are, k = = 90, and k =. 6 Algebra Copyright Big Ideas Learning, LLC
6. a + a + a + a = 60 6a = 60 6a 6 = 60 6 a = 60 So, a = 60 and the measures of the angles of the quadrilateral are a = 60, a = 60 = 0, a = 60, and a = 60 = 0. 7. (b 90) + b + b + (b + ) + 90 = 0 b + = 0 b = 9 b = 9 b = 90 So, b = 90 and the measures of the angles of the pentagon are (b 90) = 90 90 = 90, b = 90 =, b = 90, (b + ) = 90 + =, and 90. 8. x + 0 + 00 + 0 + (x + 0) + 0 = 70 x + 70 = 70 70 70 x = 0 x = 0 x = So, x = and the measures of the angles of the hexagon are x =, 0, 00, 0, (x + 0) = + 0 =, and 0. 9. n + = 7 n = 6 n = 6 n = The number is.. 8 + n = 8 8 n = 0 n = ( 0) n = 0 The number is 0.. n + n = 0 ( + ) n = 0 n = 0 n = 0 n = The number is.. 6(n + ) = 6(n) + 6() = 6n + 90 = 90 90 6n = 6n 6 = 6 n = The number is.. (n 7) = (n) (7) = n 8 = + 8 + 8 n = 0 n = 0 n = 0 The number is 0. 0. n = 9 + + n = n = n = The number is. Copyright Big Ideas Learning, LLC Algebra 7
. Words: Total earnings = Gas station hours worked Gas station hourly + wage Landscaper hourly wage Landscaper hours worked Variable: Let t be the hours you must work as a landscaper. Equation: 00 = 0(8.7) + t 00 = 0(8.7) + t 00 = 6. + t 6. 6. 7. = t 7. = t. = t Check: dollars =? hours dollars hour dollars =? dollars + dollars dollars = dollars + dollars hour hours You must work. hours as a landscaper to earn $00 per week. 6. Words: Area of swimming pool = surface Length of deep end Width of deep end Variable: Let d be the length of the deep end. + Length Width of of shallow shallow end end Equation: 0 = d 0 + 9 0 0 = 0d + 90 90 90 0 = 0d 0 0 = 0d 0 = d Check: square feet =? feet feet + feet feet square feet =? square feet + square feet square feet = square feet The deep end is feet long. 7. Words: Total cost 8 Algebra Copyright Big Ideas Learning, LLC = Sales tax as a decimal Cost of salad ( + Cost of salad Cost of one taco + + Variable: Let t be the cost of one taco. ) Cost of + Tip one taco Equation:.80 =. + t + 0.08 (. + t) +.8 =. + t + 0.08(. + t) +.8 =. + t + 0. + 0.6t +.8 =.6t +.7.7.7 8. =.6t 8..6 =.6t.6.7 = t Check: dollars =? dollars + dollars + %(dollars + dollars) + dollars dollars =? dollars + dollars + dollars + dollars + dollars dollars = dollars The cost of one taco is $.7. 8. (x 8) = 6 Write the equation. (x 8) = 7 Add to each side. x 8 = Multiply each side by. x = 6 x = 6 Add 8 to each side. Divide each side by. 9. (x + ) + x = 9 Write the equation. (x) + () + x = 9 x + 6 + x = 9 x + 6 = 9 x = Distributive Property Simplify. Combine like terms. Subtract 6 from each side. x = Divide each side by. 0. The negative sign was not distributed correctly to each term inside the parentheses. (7 y) + = (7) ( y) + = + y + = 0 + y = + 0 + 0 y = 6 y = 6 y = The solution is y =.
. In order to undo multiplying by, you should divide each side by, or multiply each side by. (x ) + = (x ) = 8 (x ) = 8 x = + + x = The solution is x =.. P = + w 8 = (w + 6) + w 8 = (w) + (6) + w 8 = w + + w 8 = 6w + 6 = 6w 6 6 = 6w 6 6 = w w + 6 = 6 + 6 = 78 The court is 78 feet by 6 feet.. P = + w 90 = ( 8 y ) + (y) 90 = y + y 90 = ( + 8 90 = 9 y ) y 9 90 = 9 9 y 0 = y 8 y = 8 0 = The Norwegian flag is inches by 0 inches.. P = s + (s + 6) + (s + 6) + s + s 0 = 6s + 90 = 6s 90 6 = 6s 6 = s s + 6 = + 6 = s = = 0 The school crossing sign has two sides that are each inches, two sides that are each inches, and one side that is 0 inches.. a. ( 8x) + 6 = () (8x) + 6 = 8 6x + 6 = 6x = 6x = 6x 6 = 6 x = 6 b. ( 8x) + 6 = 6 6 ( 8x) = 7 ( 8x) = 7 8x = 7 8x = 8 8x = x = 6 The solution is x = 6. 8 ( ) Sample answer: Method is preferred because it requires fewer operations with fractions and was therefore less complicated, making mistakes less likely. Copyright Big Ideas Learning, LLC Algebra 9
6. Words: Total cost = ( Ticket price + Convenience charge Number ) of tickets + Processing charge Variable: Let t be the number of tickets purchased. Equation: 0.70 = (.0 +.0) t +.90 0.7 = (. +.)t +.9 0.7 =.8t +.9.9.9.8 =.8t.8.8 =.8t.8 6 = t For an order that costs $0.70, 6 tickets are purchased. 7. Words Value Total value = per Number Value of + per Number of dime dimes quarter quarters Variable: Let d be the number of dimes. Equation:.80 = 0.0 d + 0. (d + 8).8 = 0.d + 0.(d + 8).8 = 0.d + 0.d +.8 = 0.d + 0.8 = 0.d 0.8 0. = 0.d 0..9 d no; Sample answer: Because it is not possible to have a decimal number of dimes, it is not possible for the number of quarters to be 8 more than the number of dimes when the total of the dimes and quarters is $.80. 8. Sample answer: Component Class Participation Student s score Weight Score weight 9% 0.0 9% 0.0 = 8.% Homework 9% 0.0 9% 0.0 = 9% Midterm Exam 88% 0. 88% 0. = % Final Exam f 0. f 0. = 0.f Total 8. + 9 + + 0.f 8. + 9 + + 0.f = 90 9. + 0.f = 90 9. 9. 0.f = 0.6 0.f 0. = 0.6 0. f 87. The student must earn at least an 88% on the final exam in order to earn an A in the class. 9. Let n be an integer. Then n is an even integer. The next even integer is more than n: n +. The third even integer is more than n + : n + + = n +. So, the total is n + (n + ) + (n + ). n + (n + ) + (n + ) = n = 8 = 6 n + n + + n + = n + = 8 + = 8 n + = 8 + = 0 6n + 6 = 6 6 6n = 8 6n 6 = 8 6 n = 8 So, the consecutive even integers are 6, 8, and 0. 0 Algebra Copyright Big Ideas Learning, LLC
0. a. greater than; Only one of the numbers is greater than 0, and two are less than 0. In order for the average to be 0, the fourth meeting must have had more than 0 students in attendance. b. Sample answer: about students c. Sample answer: You can write and solve an equation to find how many students were in attendance at the fourth meeting. If you let x be the number of students in attendance at the fourth meeting, the equation is 8 + + 7 + x = 0. 8 + + 7 + x = 0 8 + + 7 + x = 0 8 + + 7 + x = 80 6 + x = 80 6 6 x = There were students at the fourth meeting. This is close to the estimate in part (b).. bx = 7 bx b = 7 b x = 7 b. x + a = a a x = a. ax b =. + b + b ax =. + b ax a =. + b a x =. + b a. ax + b = c b b ax = c b ax a = c b a x = c b a. bx bx = 8 x(b b) = 8 x(b) = 8 x(b) b = 8 b x = 8 b 6. cx b = b + b + b cx = 9b cx c = 9b c x = 9b c Maintaining Mathematical Proficiency 7. m + m = m m + = m + 8. 9 8b + 6b = 9 + ( 8b) + 6b = 9 + ( b) = 9 b 9. 6t + ( t) = 6t + () (t) = 6t + 6t = 6t 6t + = 0 = 60. a. x 8 = 9 b. x 8 = 9 8 =? 9 9 = 9 x = is a solution. 8 =? 9 6 9 x = is not a solution. 6. a. x +. =. b. x +. =. +. =?. +. =?. 0... =. x = is not a solution. x = is a solution. 6. a. x = b. x = ( ) =? () =? =? =? x = is not a solution. = x = is a solution. Copyright Big Ideas Learning, LLC Algebra
6. a. x + = b. x + = ( ) + =? () + =? + =? 6 + =? = 0 x = is a solution. x = is not a solution. 6. a. x + = x b. x + = x + =? ( ) + =? () 6 = 6 x = is not a solution. x = is a solution. 6. a. (x ) = x b. (x ) = x ( ) =? ( ) ( ) =? () ( ) =? + () =? 6 x = is a solution.. Explorations (p. 9) = x = is not a solution.. x + + + x + + = + x + + + ( ) + ( ) x x The solution is x =. x + = x + 0 x = x + 0 0 0 = x Sample answer: Add the side lengths of each polygon to get the perimeters, set them equal to each other, and solve for x. The perimeter of the first polygon is + + + + + =. The perimeter of the second polygon is also + ( () + + + + = + 6 + + + + =. ). a. + + + x + = x + x x + 8 = x + x x + 8 = x x x 8 = 9 x 9 8 = 9 9 x = x The solution is x =. Sample answer: Add the side lengths to get the perimeter. Add the area of the triangle to the area of the rectangle to get the total area. Then set the perimeter equal to the area and solve for x. The perimeter is + + + + = feet. The area is + b. 6 + x + 6 + x + + = 6x () x The solution is x =. x + = 6x = 6 + 6 = square feet. x = x + + 6 = x 6 = x = x Sample answer: Add the side lengths to get the perimeter. Subtract the area of the small rectangle from the area of the large rectangle to get the total area. Then set the perimeter equal to the area and solve for x. The perimeter of the figure is 6 + + 6 + + + = feet. The area of the figure is 6() () = = square feet. c. π + x + + x = π + x x x + π + = π + x π The solution is x =. x π + = π + x π = x = x = x Sample answer: Add the circumference of the semicircle to the remaining three side lengths to find the perimeter. Add the area of the semicircle to the area of the rectangle to find the total area. Then set the perimeter equal to the area and solve for x. The perimeter of the figure is + + + = π + 8 feet. π The area of the figure is π + = π + 8 square feet. Algebra Copyright Big Ideas Learning, LLC
. To solve an equation that has variables on both sides, collect the variable terms on one side of the equation and the constant terms on the other side of the equation, then solve.. Sample answer: Some sample equations are x = x + {x = }, x 7 = 7x {x = }, and (x ) + = x + {x = }. Note that students may write equations with no solution or infinitely many solutions.. Monitoring Progress (pp. 0 ). x = x + 0. x x = 0 x x = 0 x = Check: x = x + 0 ( ) =? ( ) + 0 =? 6 + 0 = The solution is x =. (6h ) = h + (6h) () = h + h = h + h h = 8h + = 8h 8 = 8h 8 8 = h Check: (6h ) = h + 8) ] =? ( ( 9 ) =? 8 + [ 6 ( 8) + ( 9 6 ) =? 8 + 8 8 ( 7 = )? 7 8 7 8 = 7 8 The solution is h = 8.. (8n + ) = (n ) (8n) () = (n) () 6n 9 = n 9 + 6n + 6n 9 = 9n 9 + 9 + 9 0 = 9n 0 9 = 9n 9 0 = n Check: (8n + ) = (n ) (8(0) + ) =? (0 ) (0 + ) =? ( ) () =? 9 9 = 9 The solution is n = 0.. ( p) = p + () (p) = p + p = p + + p + p = The statement = is always true. So, the equation is an identity and has infinitely many solutions.. 6m m = m = (6m 0) 6 6 (6m) 6 (0) m = m m m 0 = The statement 0 = no solution. 6. 0k + 7 = 0k + 0k + 0k 0k + 7 = 7 7 0k = 0 0k 0 = 0 0 k = The solution is k =. is never true. So, the equation has Copyright Big Ideas Learning, LLC Algebra
7. (a ) = (a ) (a) () = (a) () 6a 6a 6 = 6a 6 6a 6 = 6 The statement 6 = 6 is always true. So, the equation is an identity and has infinitely many solutions. 8. Words: Distance upstream = Distance downstream Variable: Let x be the speed (in miles per hour) of the boat upstream. Equation: x mi h.x =.() = 7.. h = (x + ) mi h. h.x =.(x + ).x =.x +.x.x x = x = The boat travels 7. miles upstream.. Exercises (pp. ) Vocabulary and Core Concept Check. ( x) = x + 8 () ( x) = x + 8 8 + x = x + 8 x 8 = 8 x no; The equation gives a statement that is never true, so the equation has no solution and is not an identity.. (x 8) = x + 6 (x) (8) = x + 6 x 9x = x + 6 x x = 6 + + x = 0 x = 0 x = 6 Sample answer: To solve (x 8) = x + 6, the first step is to use the Distributive Property on the left side of the equal sign to remove the parentheses and then simplify so that the equation becomes 9x = x + 6. In order to eliminate the x-term from the right side, subtract x from each side. Then, in order to isolate the remaining x-term, undo subtraction by adding to each side. Finally, you can solve for x by undoing multiplication, so divide each side by. The solution is x = 6. Monitoring Progress and Modeling with Mathematics. x = x + x + x = x = x = x Check: x = x () =? () 6 =? 9 9 = 9 The solution is x =.. 6 s = 9s + s + s 6 = s 6 = s = s Check: 6 s = 9s 6 () =? 9() 6 8 =? 8 8 = 8 The solution is s =.. p 9 = p + p p p 9 = + 9 + 9 p = p = p = 7 Check: p 9 = p + (7) 9 =? (7) + 9 =? + 6 = 6 The solution is p = 7. Algebra Copyright Big Ideas Learning, LLC
6. 8g + 0 = + g g g + 0 = 0 0 g = g = g = g Check: 8g + 0 = + g 8() + 0 =? + () 0 + 0 =? + 0 = 0 The solution is g =. 7. t + 6 = 6 t + t + t 0t + 6 = 6 6 6 0t = 0 0t 0 = 0 0 t = Check: t + 6 = 6 t ( ) + 6 =? 6 ( ) + 6 =? 6 + = The solution is t =. 8. r + 0 = r 8 + r + r 0 = 8r 8 + 8 + 8 8 = 8r 8 8 = 8r 8 = r Check: r + 0 = r 8 () + 0 =? () 8 + 0 =? 8 7 = 7 The solution is r =. 9. 7 + x x = x + 7 9x = x + + 9x + 9x 7 = x + 6 = x 6 = x = x Check: 7 + x x = x + 7 + ( ) ( ) =? ( ) + 7 + 6 =? + + =? + 7 =? The solution is x =. = 0. w + w = 6 + w w = w = 6 6 8 = w 8 = w = w 6 + w w 6 + w Check: w + w = 6 + w + ( ) =? 6 + ( ) 8 =? 6 0 6 8 =? 6 0 = The solution is w =. Copyright Big Ideas Learning, LLC Algebra
. 0(g + ) = (g + 9) 0(g) + 0() = (g) + (9) 0g + 0 = g + 8 g g 8g + 0 = 8 0 = 0 8g = 8g 8 = 8 g = Check: 0(g + ) = (g + 9) 0( + ) =? ( + 9) 0() =? () 0 = 0 The solution is g =.. 9(t ) = (t ) 9(t) 9( ) = (t) () 9t + 8 = t 60 + 9t + 9t 8 = t 60 + 60 + 60 78 = t 78 = t 6 = t Check: 9(t ) = (t ) 9(6 ) =? (6 ) 9() =? ( 9) The solution is t = 6. 6 = 6. (x + 9) = (x + 6) (x) + (9) = (x) (6) x + 6 = x + x + x 6x + 6 = 6 6 6x = 8 6x 6 = 8 6 x = Check: (x + 9) = (x + 6) [ ( ) + 9 ] =? [ ( ) + 6 ] ( 9 + 9) =? ( 6 + 6) (0) =? (0) 0 = 0 The solution is x =.. (t + ) = ( 8t) (t) + () = () (8t) t + 8 = 8 6t + 6t + 6t 0t +8 = 8 8 8 0t = 0 0t 0 = 0 0 t = Check: (t + ) = ( 8t) [ () + ] =? [ 8() ] ( + ) =? ( 8) (6) =? (6) = The solution is t =. 6 Algebra Copyright Big Ideas Learning, LLC
. 0(y + ) y = (8y 8) 0(y) + 0() y = (8y) (8) 0y + 0 y = 6y 6 9y + 0 = 6y 6 6y 6y y + 0 = 6 0 0 y = 6 y = 6 y = Check: 0(y + ) y = (8y 8) 0 [ ( ) + ] ( ) =? [ 8( ) 8 ] 0( + ) + =? ( 96 8) 0( ) + =? ( 0) The solution is y =. 0 + =? 08 08 = 08 6. (x + ) = x (x ) (x) + () = x (x) ( ) 8x + = x x + 8x + = 8x + + 8x + 8x 6x + = 6x = 8 6x 6 = 8 6 x = Check: (x + ) = x (x ) [ ( ) ] + =? ( ) ( ( + ) =? ( ) () =? + 6 8 = 8 The solution is x =. ) 7. 0h = 90 h + h + h 9h = 90 9h 9 = 90 9 h = You and your friend will meet after you have been driving toward each other for hours. 8..r + =.r.r.r = 0.7r 0.7 = 0.7r 0.7 0 = r You must rent 0 movies to spend the same amount at each store. 9. t + = + t t = t The statement = is never true. So, the equation has no solution. 0. 6d + 8 = + d d d + 8 = 8 8 d = 6 d = 6 d = d The equation has one solution: d =.. (h + ) = h 7 (h) + () = h 7 h h + = h 7 h = h 7 + 7 + 7 9 = h 9 = h = h The equation has one solution: h =. Copyright Big Ideas Learning, LLC Algebra 7
. y + 6 = 6(y + ) y + 6 = 6(y) + 6() y + 6 = y + 6 y 6 = 6 y The statement 6 = 6 is always true. So, the equation is an identity and has infinitely many solutions.. (g + 6) = (6g + 9) (g) + (6) = (6g) + (9) g + 8 = g + 8 g 8 = 8 g The statement 8 = 8 is always true. So, the equation is an identity and has infinitely many solutions.. ( + m) = (8 + 0m) () + (m) = (8) + (0m) + 0m = + 0m 0m = 0m The statement = is never true. So, the equation has no solution.. In order to undo subtraction, c should have been added to each side. c 6 = c + c + c 8c 6 = + 6 + 6 8c = 0 8c 8 = 0 8 c = The solution is c =. 6. Because the statement 0 = 0 is always true, the equation has infinitely many solutions. It is better to subtract a variable term from each side before subtracting a constant from each side. 6(y + 6) = (9 + y) 6(y) + 6(6) = (9) + (y) y + 6 = 6 + y y 6 = 6 y Because the statement 6 = 6 is always true, the equation has infinitely many solutions. 7. Words: Total cost of Company A s Internet service = Total cost of Company B s Internet service Variable: Let m be the number of months you have Internet service. Equation: 60.00 +.9m =.00 + 9.9m 60 +.9m = + 9.9m.9m.9m 60 = + 7m = 7m 7 = 7m 7 = m After months, you would pay the same total amount for each Internet service. 8. Words: % of total protein needed daily + 8 = Total protein needed daily Variable: Let p be the total amount (in grams) of protein you need daily. Equation: 0.0 p + 8 = p 0.0p + 8 = p 0.0p 0.0p 8 = 0.96p 8 0.96 = 0.96p 0.96 0 = p You need 0 grams of protein daily. 9. 8(x + 6) 0 + r = (x + ) + x 8(x) + 8(6) 0 + r = (x) + () + x 8x + 8 0 + r = x + 6 + x 8x So, r =. 8x + 8 + r = 8x + 6 8x 8 + r = 6 8 8 r = 8 Algebra Copyright Big Ideas Learning, LLC
0. (x ) r + x = (x 7) 9x (x) () r + x = (x) (7) 9x x r + x = x 9x 6x So, r =. 6x r = 6x 6x r = + + r = r = r =. πr + πrh = πr h π(.) + π(.)(x) = π(.) (x) SA = πr + πrh.π + πx = 6.πx πx πx.π =.πx.π.π =.πx.π 0 = x V = πr h = π(.) + π(.)(0) = π(.) (0) =.π + 0π = π(6.)(0) = 6.π = 6.π 6.(.6) 6.(.6) = 96. = 96. So, x = 0, and the surface area is 6.π, or about 96. square centimeters and the volume is 6.π, or about 96. cubic centimeters.. πr + πrh = πr h π ( 8 ) + π ( 8 ) (x) = π ( 8 ) (x) π ( ) + 6 πx = πx 68 π + 6 πx = πx 6 πx 6 πx 68 π = πx 68 π = πx SA = πr + πrh = π ( 8 9 π = πx 9 π π = πx π 9 = x ) + π ( 8 = π ( ) + π ( 8 V = πr h ) ( 9 ) = π ( 8 ) ( 9 ) ) = π ( ) ( 9 ) = 68 π + 6 π = 8 π = 68 π + 80 π 8 (.6) = 8 π 8. 8 (.6) 8. So, x = 9, and the surface area is 8 π, or about 8. square feet and the volume is 8 π, or about 8. cubic feet.. Words: Cheetah s running distance = 0 feet + Antelope s running distance Variable: Let t be the time (in seconds) the animals are running. 90 ft 60 ft Equation: sec t sec = 0 ft + sec t sec 90 t = 0 + 60 t t = The cheetah will catch up to the antelope in seconds. Copyright Big Ideas Learning, LLC Algebra 9 60 t 0 t = 0 0 t 0 = 0 0 60 t
. 90 ft 60 ft sec t sec = 60 ft + sec t sec 90t = 60 + 60t 60t 0t = 60 0t 0 = 60 0 60t t = sec no; In order to catch the antelope, the cheetah would have to be running at top speed for over 0 seconds, so the antelope is probably safe.. a(x+) = 9x + + x a(x) + a() = 0x + ax + a = 0x + Set the coefficients of x equal to each other and set the constant terms equal to each other. a = 0 a = a = 0 a = a = a = If a =, then ax + a = () x + () = 0x +. Because the other side of the equal sign is also 0x +, the equation is an identity when a =. 6. 8x 8 + ax = ax a 8x + ax 8 = ax a (8 + a)x 8 = ax a Set the coefficients of x equal to each other and set the constant terms equal to each other. 8 + a = a 8 = a 8 a a = a 8 = a = a 8 = a = a If a =, then 8x 8 + ax = 8x 8 + ()x = 8x 8 + x = 0x 8, and ax a = ()x () = 0x 8. Because the expressions on each side of the equation are the same, the equation is an identity when a =. 7. Words: Greater consecutive integer = Lesser consecutive integer 9 Variable: Let n be the lesser consecutive integer. Then n + is the greater consecutive integer. Equation: (n + ) = n 9 (n + ) = n 9 (n) + () = n 9 n n + = n 9 n = n 9 + 9 + 9 = n n + = + = The integers are and. 8. a. After 6 years, there will be equal enrollment in Spanish and French classes because the graphs meet at this point. b. The left side, 9x, represents the predicted Spanish class enrollment, and the right side, 9 + x, represents the predicted French class enrollment. So, the equation represents when there will be equal enrollment in Spanish and French classes. The solution should give the same result as the graph. 9x = 9 + x + 9x + 9x = 9 + x 9 9 6 = x 6 = x 6 = x The equation confirms that there will be equal enrollment in Spanish and French classes after 6 years if the trend continues as predicted. 0 Algebra Copyright Big Ideas Learning, LLC
9. a. Sample answer: (x + ) = x 7 (x) + () = x 7 x x + 0 = x 7 x 0 = 7 The statement 0 = 7 is never true. So, the equation (x + ) = x 7 has no solution. b. Sample answer: (x + 6) = (x + ) (x)+(6) = (x) + () 6x 6x + = 6x + 6x = The statement = is always true. So, the equation (x + 6) = (x + ) is an identity and has infinitely many solutions. 0. Sample answer: The perimeter of the given triangle is P = (x + ) + (x + ) + x = 6x +. Another figure with the same perimeter is shown. x x P = x + + x + = 6x + Maintaining Mathematical Proficiency., =, =,, 9. =, 6, 0 = 0,, =. 9, 8, 8 = 8, =, =. =,,, 0 = 0, =.. What Did You Learn? (p. ). Sample answer: Let A represent the area of a single rectangle. Because each rectangle has the same area, and the area of the square is half the area of a rectangle, use the expression A + A to represent the total area.. Sample answer: A protractor can only measure angles to the nearest whole degree. It is not a very precise instrument.. Sample answer: the definition of an identity, which has infinitely many solutions.. Quiz (p. 6). x + 9 = 7 Write the equation. 9 9 Subtract 9 from each side. x = Check: x + 9 = 7 + 9 =? 7 7 = 7 Simplify. The solution is x =.. 8.6 = z.8 Write the equation. +.8 +.8 Add.8 to each side.. = z Simplify. Check: 8.6 = z.8 8.6 =?..8 8.6 =? 8.6 The solution is z =... 60 = r Write the equation. 60 = r Divide each side by. = r Check: 60 = r 60 =? ( ) 60 = 60 The solution is r =. Simplify.. p = 8 Write the equation. p = 8 Multiply each side by. p = Simplify. Check: p = 8 () =? 8 8 = 8 The solution is p =.. m = + + m = 6 m = 6 m = 8 Check: m = (8) =? 6 =? = The solution is m = 8. Copyright Big Ideas Learning, LLC Algebra
6. = 0 v Check: = 0 v 0 0 = v = v = v The solution is v =. 7. = 7w + 8w + = w + = w = w = w Check: = 7w + 8w + =? 7 ( ) ( + 8 ) + =? 7 + 8 + =? + =? + = The solution is w =. 8. a + 8a 6 = 0. 7a 6 = 0. + 6 + 6 7a =. 7a 7 =. 7 a = 0.6 =? 0 = Check: a + 8a 6 = 0. ( 0.6) + 8( 0.6) 6 =? 0..6 6.8 6 =? 0. The solution is a = 0.6.. 6 =? 0. 0. = 0. 9. k (k ) = k (k) ( ) = k 6k + 9 = k + 9 = 9 9 k = 6 k = 6 k = 9 Check: k (k ) = ( 9) [( 9) ] =? 8 ( 8 ) =? 8 ( ) =? 8 + 6 =? = The solution is k = 9. 0. 68 = (0x + 0) + 68 = (0x) + (0) + 68 = x + 0 + 68 = x + 6 = x 6 = x = x Check: 68 = (0x + 0) + 68 =? [0() + 0] + 68 =? (80 + 0) + 68 =? (0) + 68 =? 66 + 68 = 68 The solution is x =.. c + = c + c c c + = c = 0 c 0 = c = 0 The solution is c = 0. Algebra Copyright Big Ideas Learning, LLC
. 8 n = 6 + n + n + n 8 = 6 6 7 = 8n 7 8 = 8n 8 9 = n 6 + 8n The solution is n = 9.. (8q ) = q (8q) () = 6q 0 = 6q q q 6q 0 = q 0 = q 6 = q The solution is q = 6.. 9(y ) 7y = (y ) 9(y) 9() 7y = (y) () 9y 6 7y = y 0 y y 6 = y 0 y 6 = y 0 + 0 + 0 6 = y 6 = y = y The solution is y =. 6. ( h ) = (0h + 8) ( h) ( ) = (0h) + (8) 0h + 6 = 0h + 6 0h 6 = 6 0h The statement 6 = 6 is always true. So, the equation is an identity and has infinitely many solutions. 7. Words: Distance from a thunderstorm Variable: = Time between lightning and thunder Let s be the time (in seconds) between when you see lightning and when you hear thunder. Equation: = s = s = s 0 = s You would count 0 seconds between when you see lightning and when you hear thunder for a thunderstorm that is miles away. 8. Let x be the spacing (in feet) between the posters. = + + x + + x + + = x + = x = x = x There should be or feet between the posters.. (g + 8) = 7 + g (g) + (8) = 7 + g g g + = 7 + g = 7 g The statement = 7 is never true. So, the equation has no solution. Copyright Big Ideas Learning, LLC Algebra
9. a. Words: Total cost for Studio A = Total cost for Studio B Variable: Let h be the time (in hours) spent painting at the studio. Equation: 0 + 8h = 6 + 6h 0 + 8h = 6 + 6h 6h 0 + h = 6 0 0 h = 6 h = 6 h = 6h After hours of painting, the total costs will be the same at both studios. b. If Studio B increases their studio fee by $, then both studios have the same studio fee of $8. 0 + 8h = 6 + 8h 8h 0 = 6 8h The statement 0 = 6 is never true. So, the equation has no solution. In other words, because Studio B charges more for the vase, if their studio fee is the same, their total costs will never be the same. More specifically, Studio B will always charge more.. Explorations (p. 7). a. Because = and =, it must be that x + equals either or. So, x + = or x + =. b. x + = or x + = x = or x = The solutions are x = and x =. c. Sample answer: If an absolute value expression is equal to a constant, then the expression is equal to the constant or its opposite. You can write two linear equations, one for each of these possibilities, and then solve both equations.. a. b. 8 6 x + = 0 + =? 0 8 0 = 0 6 0 0 Both of these values, and, are solutions of the absolute value equation x + =. x + = x + = + =? + =? =? =? = = c. Set the expression inside the absolute value symbol equal to 0 and solve. Plot the solution on a number line. Then plot the points that are the constant amount of units from that point. These last points are the solutions to the original equation.. a. A x -6 - - - 6-7 - 8 0 9 0 B x + 0 The solutions given by the spreadsheet are and, because they are the values of x that make x + equal to. b. The spreadsheet method yielded the same solutions, and, as the other two methods. c. Sample answer: Have the spreadsheet calculate the value of the absolute value expression for many values of x, and find the ones that give the expected solution.. Sample answer: You can solve an absolute value equation algebraically, graphically, or numerically. For the algebraic method, write and solve two linear equations, one that has the expression equal to the constant and one that has the expression equal to the opposite of the constant. For the graphical method, first identify the point on the number line that makes the absolute value expression equal to 0. Using the constant that the absolute value expression is equal to, find both of the points that are this many units from the original point in either direction. These two values are the solutions. For the numerical method, you can use a spreadsheet to calculate the values of the absolute value expression for many values of the variable until you identify the value or values that make the absolute value expression equal to the given constant. Algebra Copyright Big Ideas Learning, LLC
. Sample answer: The algebraic method is favorable because it is the quickest method. The graphical method is also favorable because it helps to visualize absolute value. The numerical method is not favorable because setting up the spreadsheet is time consuming.. Monitoring Progress (pp. 8 ). x = 0 x = 0 or x = 0 0 8 0 0 8 The solutions are x = 0 and x =0.. x = x = or x = + + + + x = 0 6 The solutions are x = and x =.. + x = x = The absolute value of an expression must be greater than or equal to 0. The expression + x cannot equal. So, the equation has no solution.. x + = 9 x = x = or x = + + + + x = 6 x = Check: x + = 9 x + = 9 6 + =? 9 + =? 9 + =? 9 + =? 9 + =? 9 + =? 9 9 = 9 9 = 9 The solutions are x = and x = 6.. x + 7 = 6 x + 7 = 6 x + 7 = x + 7 = or x + 7 = Check: 7 7 7 7 x = x = x = x = x = x = x + 7 = 6 x + 7 = 6 ( + 7 ) =? 6 ( ) + 7 =? 6 + 7 =? 6 + 7 =? 6 =? 6 =? 6 =? 6 =? 6 6 = 6 6 = 6 The solutions are x = and x =. 6. x = + + x = 8 x = 8 x = x = or x = + + + + x = x = x = x = x = x = Check: x = x = () =? ( ) =? =? =? () =? 8 =? = =? =? () =? 8 =? = The solutions are x = and x =. Copyright Big Ideas Learning, LLC Algebra
7. 8 8 6 0 8 6 The equation is x = 8. Check: x = 8 x = 8 6 =? 8 =? 8 8. x + 8 = x + 8 =? 8 8 =? 8 8 = 8 8 = 8 x + 8 = x + or x + 8 = (x + ) x x x + 8 = x 8 = x + + x + x x + 8 = 7 = x 8 8 Check: x + 8 = x + 7 + 8 =? (7) + =? + x = 9 x = 9 x = x + 8 = x + + 8 =? ( ) + =? 6 + =? =? = = The solutions are x = and x = 7. 9. x = x + (x ) = x + or (x ) = (x + ) (x) () = x + (x) () = x x = x + x = x x x + x + x x = x = + + + + x = 7 x = 7 x = 7 x = 7 Check: x = x + x = x + 7 =? (7) + 7 =? ( 7 ) + =? + 7 0 =? () =? 9 =? 9 = 9 ( ) =? 9 The solutions are x = 7 and x = 7. 0. x + 6 = x x + 6 = x or x + 6 = x x x x x 6 = x 6 = x 6 = x = x Check: x + 6 = x 6 + 6 =? (6) =? 9 = 9 x + 6 = x + 6 =? ( ) =? = + + The solution is x = 6. Reject x = because it is extraneous.. x = x x = x or x = x x x x x = x = x = x Check: x = x = x = x = x x = x () =? ( ) =? =? =? =? =? = =? The solutions are x = and x =. = 6 Algebra Copyright Big Ideas Learning, LLC
. + x = x 8 + x = x 8 or + x = (x 8) x x + x = x + 8 = 8 + x + x The statement = 8 is false. So, the original equation has only one solution. Check: + x = x 8 + =? 8 =? = The solution is x =.. x = x + + x = 8 x = 6 x = 6 x = x = x + or x = (x + ) x x x = x = + x + x The statement = is false. So, the original equation has only one solution. Check: x = x + ( ) =? ( + ) =? + =? = The solution is x =.. Exercises (pp. ) Vocabulary and Core Concept Check 0x = + + 0x = 0x 0 = 0 x =. An extraneous solution is an apparent solution that must be rejected because it does not satisfy the original equation.. The absolute value of an expression must be greater than or equal to 0. So, the expression x 7 cannot equal, and the equation has no solution. Monitoring Progress and Modeling with Mathematics. 9 = 9. =. = = 0 6. + = + = 6 7. ( 7) = = 8. 0.8 0 = 8 = 8 9. 7 = 9 = 9 0. = ( ) = =. w = 6 w = 6 or w = 6 6 0 6 The solutions are w = 6 and w = 6.. r = The absolute value of a number must be greater than or equal to 0 and cannot be equal to. So, the equation has no solution.. y = 8 The absolute value of a number must be greater than or equal to 0 and cannot be equal to 8. So, the equation has no solution.. x = x = or x = 8 0 8 The solutions are x = and x =.. m + = 7 m + = 7 or m + = 7 0 m = m = 0 8 6 0 The solutions are m = 0 and m =. 6. q 8 = q 8 = or q 8 = + 8 + 8 + 8 + 8 6 q = q = 6 8 0 8 6 The solutions are q = 6 and q =. Copyright Big Ideas Learning, LLC Algebra 7