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Welcome to Interactive Chalkboard Algebra 1 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION.

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Presentation on theme: "Welcome to Interactive Chalkboard Algebra 1 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION."— Presentation transcript:

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2 Welcome to Interactive Chalkboard Algebra 1 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

3 Splash Screen

4 Contents Lesson 12-1Inverse Variation Lesson 12-2Rational Expressions Lesson 12-3Multiplying Rational Expressions Lesson 12-4Dividing Rational Expressions Lesson 12-5Dividing Polynomials Lesson 12-6Rational Expressions with Like Denominators Lesson 12-7Rational Expressions with Unlike Denominators Lesson 12-8Mixed Expressions and Complex Fractions Lesson 12-9Solving Rational Equations

5 Lesson 1 Contents Example 1Graph an Inverse Variation Example 2Graph an Inverse Variation Example 3Solve for x Example 4Solve for y Example 5Use Inverse Variation to Solve a Problem

6 Example 1-1a Manufacturing The owner of Superfast Computer Company has calculated that the time t in hours that it takes to build a particular model of computer varies inversely with the number of people p working on the computer. The equationcan be used to represent the people building a computer. Complete the table and draw a graph of the relation. p t

7 Example 1-1b p t Original equation Replace p with 2. Divide each side by 2. Simplify. 6 Solve the equation for the other values of p Answer: Solve for

8 Example 1-1c Answer: Graph the ordered pairs: (2, 6), (4, 3), (6, 2), (8, 1.5), (10, 1.2), and (12, 1). As the number of people p increases, the time t it takes to build a computer decreases.

9 p t p t Example 1-1d Answer: Manufacturing The foreman of a package delivery company has found that the time t in hours that it takes to prepare packages for delivery varies inversely with the number of people p that are preparing them. The equationcan be used to represent the people preparing the packages. Complete the table and draw a graph of the relation.

10 Example 1-1e Answer:

11 Example 1-2a Inverse variation equation The constant of variation is 4. Graph an inverse variation in which y varies inversely as x and Solve for k.

12 Example 1-2b xy –4–1 –2 –1–4 0 undefined Choose values for x and y whose product is 4.

13 Example 1-2c Answer: Graph an inverse variation in which y varies inversely as x and

14 Example 1-3a Method 1Use the product rule. Product rule for inverse variations Divide each side by 15. Simplify. If y varies inversely as x and find x when

15 Example 1-3b Method 2Use a proportion. Proportion rule for inverse variations Cross multiply. Divide each side by 15. Answer:Both methods show that

16 Example 1-3c If y varies inversely as x and find x when Answer: 8

17 Example 1-4a If y varies inversely as x and find y when Use the product rule. Product rule for inverse variations Divide each side by 4. Simplify. Answer:

18 Example 1-4b If y varies inversely as x and find y when Answer: –25

19 Example 1-5a Physical Science When two objects are balanced on a lever, their distances from the fulcrum are inversely proportional to their weights. How far should a 2-kilogram weight be from the fulcrum if a 6-kilogram weight is 3.2 meters from the fulcrum?

20 Example 1-5b Original equation Divide each side by 2. Simplify. Answer:The 2-kilogram weight should be 9.6 meters from the fulcrum.

21 Example 1-5c Answer:1 m Physical Science How far should a 10-kilogram weight be from the fulcrum if a 4 kilogram weight is 2.5 meters from the fulcrum?

22 End of Lesson 1

23 Lesson 2 Contents Example 1One Excluded Value Example 2Multiple Excluded Values Example 3Use Rational Expressions Example 4Expression Involving Monomials Example 5Expression Involving Polynomials Example 6Excluded Values

24 Example 2-1a Exclude the values for which Subtract 7 from each side. Answer: b cannot equal –7. The denominator cannot equal zero. State the excluded value of

25 Example 2-1b Answer: –3 State the excluded value of

26 Example 2-2a Exclude the values for which The denominator cannot equal zero. Factor. Use the Zero Product Property to solve for a. or Answer: a cannot equal –3 or 4. State the excluded values of

27 Example 2-2b Answer: 2, 3 State the excluded values of

28 Example 2-3a The original mechanical advantage was 5. Landscaping Refer to Example 3 on page 649. Suppose Kenyi finds a rock that he cannot move with a 6-foot bar, so he gets an 8-foot bar. But this time, he places the fulcrum so that the effort arm is 6 feet long, and the resistance arm in 2 feet long. Explain whether he has more or less mechanical advantage with his new setup.

29 Example 2-3b Simplify. Answer:Even though the bar is longer, because he moved the fulcrum he has a mechanical advantage of 3, so his mechanical advantage is less than before. Use the expression for mechanical advantage to write an expression for the mechanical advantage in the new situation.

30 Example 2-3c Answer:Since the mechanical advantage is 3, Kenyi can lift or 540 pounds with the longer bar. If Kenyi can apply a force of 180 pounds, what is the greatest weight he can lift with the longer bar?

31 Example 2-3d Landscaping Sean and Travis are responsible for clearing an area for a garden. They come across a large rock that they cannot lift. Therefore, they use a 5-foot bar as a lever, and the fulcrum is 1 foot away from the rock. a. Use the formulato find the mechanical advantage. b. If they can apply a force of 200 pounds, what is the greatest weight they can lift? Answer: 4 Answer: 800 lb

32 Example 2-4a The GCF of the numerator and denominator is Divide the numerator and denominator by 1 1 Simplify Answer: Simplify.

33 Example 2-4b Simplify Answer:

34 Example 2-5a Factor. Divide the numerator and denominator by the GCF, x – Simplify Answer:Simplify

35 Example 2-5b Simplify Answer:

36 Example 2-6a Divide the numerator and denominator by the GFC, x SimplifyState the excluded values of x. Factor. Simplify.Answer:

37 Example 2-6b Exclude the values for whichequals 0. The denominator cannot equal zero. Factor. Zero Product Property

38 Example 2-6c Evaluate. Simplify. Check Verify the excluded values by substituting them into the original expression.

39 Example 2-6d Evaluate. Simplify. Answer:The expression is undefined when and Therefore,

40 Example 2-6e Answer: SimplifyState the excluded values of w.

41 End of Lesson 2

42 Lesson 3 Contents Example 1Expressions Involving Monomials Example 2Expressions Involving Polynomials Example 3Dimensional Analysis

43 Example 3-1a Method 1Divide by the greatest common factor after multiplying. Multiply the numerators. Multiply the denominators. The GCF is 98xyz. Simplify. Find

44 Example 3-1a Method 2Divide the common factors before multiplying. Multiply.Answer: Divide by common factors and z. 1 x 1 6 z 2z 2 y 3y

45 Example 3-1a Multiply.Answer: Divide by common factors and r d 2d 2 q 2q 2 r 3r 3 Find

46 Example 3-1b Answer: a. Find b. Find

47 Example 3-2a Factor the numerator. Simplify.Answer: The GCF is x 2x 2 Find

48 Example 3-2a Find Factor. The GCF is

49 Example 3-2a Multiply. Simplify.Answer:

50 Example 3-2b a.Find b.Find Answer:

51 Example 3-3a Space The velocity that a spacecraft must have in order to escape Earths gravitational pull is called the escape velocity. The escape velocity for a spacecraft leaving Earth is about 40,320 kilometers per hour. What is this speed in meters per second?

52 Answer:The escape velocity is 11,200 meters per second. Example 3-3a Simplify. Multiply.

53 Example 3-3b Aviation The speed of sound, or Mach 1, is approximately 330 meters per second at sea level. What is the speed of sound in kilometers per hour? Answer:1188 kilometers per hour

54 End of Lesson 3

55 Lesson 4 Contents Example 1Expression Involving Monomials Example 2Expression Involving Binomials Example 3Divide by a Binomial Example 4Expression Involving Polynomials Example 5Dimensional Analysis

56 Example 4-1a Find Multiply by the reciprocal of Answer: Simplify. Divide by common factors 5, 6, and x x3x3

57 Example 4-1b Find Answer:

58 Example 4-2a Find Multiply by the reciprocal of Factor

59 Example 4-2b orSimplify.Answer: The GCF is 1 1

60 Example 4-2c Find Answer:

61 Example 4-3a Find Multiply by the reciprocal of Factor

62 Example 4-3b Simplify.Answer: The GCF is 1 1

63 Example 4-3c Find Answer:

64 Example 4-4a Find Multiply by the reciprocal, Factor

65 Example 4-4b Simplify.Answer: The GCF is 1 1

66 Example 4-4c Find Answer: or (k – 10) (k – 2) 10

67 Example 4-5a Aviation In 1986, an experimental aircraft named Voyager was piloted by Jenna Yeager and Dick Rutan around the world non-stop, without refueling. The trip took exactly 9 days and covered a distance of 25,012 miles. What was the speed of the aircraft in miles per hour? Round to the nearest mile per hour. Use the formula for rate, time, and distance. Divide each side by 9 days.

68 Example 4-5b Convert days to hours. Answer:Thus, the speed of the aircraft was about 116 miles per hour.

69 Example 4-5c Aviation Suppose that Jenna Yeager and Dick Rutan wanted to complete the trip in exactly 7 days. What would be their average speed in miles per hour for the 25,012-mile trip? Answer:about 149 miles per hour

70 End of Lesson 4

71 Lesson 5 Contents Example 1Divide a Binomial by a Monomial Example 2Divide a Polynomial by a Monomial Example 3Divide a Polynomial by a Binomial Example 4Long Division Example 5Polynomial with Missing Terms

72 Example 5-1a Find Write as a rational expression. Divide each term by 2x. 2x2x Simplify each term. Simplify.Answer:

73 Example 5-1b Find Answer:

74 Example 5-2a Find Write as a rational expression. Divide each term by 3y. Simplify.Answer: 2y2y Simplify each term. y 3

75 Example 5-2b Find Answer:

76 Example 5-3a 1 1 Divide by the GCF. Find Factor the numerator. Simplify.Answer: Write as a rational expression.

77 Example 5-3b Find Answer:

78 Example 5-4a Find Step 1Divide the first term of the dividend, x 2, by the first term of the divisor, x. x Multiply x and x – 2. Subtract.

79 Example 5-4b Step 2Divide the first term of the partial dividend, 9x – 15, by the first term of the divisor, x. x+ 9 Subtract and bring down – 15. Multiply 9 and x – 2. Subtract.

80 Example 5-4c Answer:The quotient of is with a remainder of 3, which can be written as

81 Example 5-4d Find Answer:The quotient is with a remainder of 2.

82 Example 5-5a Rename the x 2 term by using a coefficient of 0. Find

83 Example 5-5b Multiply x 2 and x – 5. Subtract and bring down – 34x. Multiply 5x and x – 5. Subtract and bring down 45. Multiply –9 and x – 5. Subtract. Answer:

84 Example 5-5c Find Answer:The quotient is

85 End of Lesson 5

86 Lesson 6 Contents Example 1Numbers in Denominator Example 2Binomials in Denominator Example 3Find a Perimeter Example 4Subtract Rational Expressions Example 5Inverse Denominators

87 Example 6-1a Find The common denominator is 15. Add the numerators. Simplify. Answer: Divide by the common factor,

88 Example 6-1b Find Answer:

89 Example 6-2a Divide by the common factor, c Find The common denominator is c + 2. Factor the numerator. Simplify. Answer:

90 Example 6-2b Answer: 5 Find

91 Example 6-3a Geometry Find an expression for the perimeter of rectangle WXYZ. Perimeter formula

92 Example 6-3b The common denominator is Distributive Property Combine like terms. Factor.

93 Example 6-3c Answer:The perimeter can be represented by the expression

94 Example 6-3d Geometry Find an expression for the perimeter of rectangle PQRS. Answer: 18

95 Example 6-4a Find The common denominator is The additive inverse of is Distributive Property

96 Example 6-4b Simplify. Answer:

97 Example 6-4c FindAnswer:

98 Example 6-5a Find The denominatoris the same as or. Rewrite the second expression so that it has the same denominator as the first. Rewrite using common denominators.

99 Example 6-5b The common denominator is Simplify. Answer:

100 Example 6-5c Find Answer:

101 End of Lesson 6

102 Lesson 7 Contents Example 1LCM of Monomials Example 2LCM of Polynomials Example 3Monomial Denominators Example 4Polynomial Denominators Example 5Binomials in Denominators Example 6Polynomials in Denominators

103 Example 7-1a Find the LCM of Find the prime factors of each coefficient and variable expression. Use each prime factor the greatest number of times it appears in any of the factorizations. Answer:

104 Example 7-1b Find the LCM of Answer:

105 Example 7-2a Find the LCM of Express each polynomial in factored form. Use each factor the greatest number of times it appears. Answer:

106 Example 7-2b Find the LCM of Answer:

107 Example 7-3a Find Factor each denominator and find the LCD. Since the denominator of is already 5z, only needs to be renamed.

108 Example 7-3b Multiply by Distributive Property Add the numerators. Answer: Simplify.

109 Example 7-3c Find Answer:

110 Example 7-4a Find Factor the denominators. The LCD is

111 Example 7-4b Add the numerators. Simplify.Answer:

112 Example 7-4c Find Answer:

113 Example 7-5a Find Factor. The LCD is Add the numerators.

114 Example 7-5b Multiply. Simplify.Answer:

115 Example 7-5c Find Answer:

116 Example 7-6a Multiple-Choice Test Item Find A B C D

117 Example 7-6b Read the Test Item The expressionrepresents the difference of two rational expressions with unlike denominators. Solve the Test Item Step 1Factor each denominator and find the LCD. The LCD is

118 Example 7-6c Step 2Change each rational expression into an equivalent expression with the LCD. Then subtract.

119 Example 7-6d Answer: C

120 Example 7-6e Multiple-Choice Test Item Find A B C D Answer: C

121 End of Lesson 7

122 Lesson 8 Contents Example 1Mixed Expression to Rational Expression Example 2Complex Fraction Involving Numbers Example 3Complex Fraction Involving Monomials Example 4Complex Fraction Involving Polynomials

123 Example 8-1a Simplify The LCD is Add the numerators. Distributive Property Answer:Simplify.

124 Example 8-1b Simplify Answer:

125 Example 8-2a Baking Suppose Katelyn bought 2 pounds of chocolate chip cookie dough. If the average cookie requires ounces of dough, how many cookies would she be able to make? To find the total number of cookies, divide the amount of cookie dough by the amount of dough needed for each cookie.

126 Example 8-2b Convert pounds to ounces and divide by common units. Simplify. Express each term as an improper fraction.

127 Example 8-2c Simplify. Answer: Katelyn can make 21 cookies.

128 Example 8-2d Answer: 27 cookies Baking James bought pounds of cookie dough, and he prefers to make large cookies. If each cookie requires ounces of dough, how many cookies can he make?

129 Example 8-3a Simplify Rewrite as a division sentence. Rewrite as multiplication by the reciprocal.

130 Example 8-3b Divide by common factors a, b, and c 2. a 4a 4 c 2c 2 b 3b Simplify.Answer:

131 Example 8-3c Answer: Simplify

132 Example 8-4a Simplify The LCD of the fractions in the numerator is Simplify the numerator. The numerator contains a mixed expression. Rewrite it as a rational expression first.

133 Example 8-4b Rewrite as a division sentence. Multiply by the reciprocal of Simplify.Answer: Factor.

134 Example 8-4c Simplify Answer:

135 End of Lesson 8

136 Lesson 9 Contents Example 1Use Cross Products Example 2Use the LCD Example 3Multiple Solutions Example 4Work Problem Example 5Rate Problem Example 6No Solution Example 7Extraneous Solution

137 Example 9-1a Solve Original equation Cross multiply. Distributive Property Add –2x and 48 to each side. Answer: Divide each side by 6.

138 Example 9-1b Answer: –3 Solve

139 Example 9-2a Solve Original equation The LCD is Distributive Property

140 Example 9-2b Simplify. Add. Subtract 1 from each side. Divide each side by 6. Answer:

141 Example 9-2c Answer: 8 Solve

142 Example 9-3a Solve Distributive Property Original equation The LCD is

143 Example 9-3b Simplify. or Set equal to 0. Factor.

144 Example 9-3c CheckCheck solutions by substituting each value in the original equation.

145 Example 9-3d CheckCheck solutions by substituting each value in the original equation. Answer:The number 1 is an excluded value for x. Thus, the solution is 3.

146 Example 9-3e Solve Answer: 4, –1

147 Example 9-4a TV Installation On Saturdays, Lee helps her father install satellite TV systems. The jobs normally take Lees father about hours. But when Lee helps, the jobs only take them hours. If Lee were installing a satellite system herself, how long would the job take?

148 Example 9-4b ExploreSince it takes Lees fatherhours to install one job, he can finish of the job in one hour. The amount of work Lee can do in one hour can be represented by To determine how long it takes Lee to do the job, use the formula Lees work + her fathers work = 1 job.

149 Example 9-4c PlanThe time that both of them worked was hours. Each rate multiplied by this time results in the amount of work done by each person. SolveLeesher fatherstotal workplus workequalswork. 1

150 Example 9-4d Multiply. The LCD is 10t. Distributive Property Simplify.

151 Example 9-4e Add –6t to each side. Divide each side by 4. Answer:The job would take Lee or hours by herself. ExamineThis seems reasonable because the combined efforts of the two took longer than half of her fathers usual time.

152 Example 9-4f Driveways Shawna earns extra money by shoveling driveways. If she works alone, she can finish a large driveway in hours. If Vince helps her, they can get done in hours. If Vince were shoveling the driveway himself, how long would the job take him? Answer:hours

153 Example 9-5a Transportation The schedule for the Washington, D.C., Metrorail is shown to the right. Suppose two Red Line trains leave their stations at opposite ends of the line at exactly 2:00 P.M. One train travels between the two stations in 48 minutes and the other train takes 54 minutes. At what time do the two trains pass each other?

154 Example 9-5b Determine the rates of both trains. The total distance is 19.4 miles. Train 1 Train 2 Next, since both trains left at the same time, the time both have traveled when they pass will be the same. And since they started at opposite ends of the route, the sum of their distances is equal to the total route, 19.4 miles.

155 Example 9-5c t min Train 2 t min Train 1 d = r t tr The sum of the distances is The LCD is 432.

156 Example 9-5d Distributive Property Simplify. Add. Divide each side by Answer:The trains passed each other at about 25 minutes after they left their stations, at 2:25 P.M.

157 Example 9-5e Transportation Two cyclists are riding on a 5-mile circular bike trail. They both leave the bike trail entrance at 3:00 P.M. traveling in opposite directions. It usually takes the first cyclist one hour to complete the trail and it takes the second cyclist 50 minutes. At what time will they pass each other? Answer:3:27 P.M.

158 Example 9-6a Solve Original equation The LCD is x – 1. Distributive Property

159 Example 9-6b Simplify. Subtract 2 from each side. Answer:Since 1 is an excluded value for x, the number 1 is an extraneous solution. Thus, the equation has no solution.

160 Example 9-6c Solve Answer:no solutions

161 Example 9-7a Solve Original equation The LCD is x – Distributive Property

162 Example 9-7b Simplify. Subtract 4 from each side. Factor. or Zero Product Property Answer:The number 2 is an extraneous solution, since 2 is an excluded value for x. Thus, –2 is the solution of the equation.

163 Example 9-7c Solve Answer: –3

164 End of Lesson 9

165 Algebra1.com Explore online information about the information introduced in this chapter. Click on the Connect button to launch your browser and go to the Algebra 1 Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to

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184 Help To navigate within this Interactive Chalkboard product: Click the Forward button to go to the next slide. Click the Previous button to return to the previous slide. Click the Section Back button to return to the beginning of the lesson you are working on. If you accessed a feature, this button will return you to the slide from where you accessed the feature. Click the Main Menu button to return to the presentation main menu. Click the Help button to access this screen. Click the Exit button or press the Escape key [Esc] to end the current slide show. Click the Extra Examples button to access additional examples on the Internet. Click the 5-Minute Check button to access the specific 5-Minute Check transparency that corresponds to each lesson.

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