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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 Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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Splash Screen

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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

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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

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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

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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

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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.

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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.

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Example 1-1e Answer:

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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.

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

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

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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

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

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

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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:

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

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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?

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

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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?

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End of Lesson 1

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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

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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

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

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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

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

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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.

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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.

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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?

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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

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

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Example 2-4b Simplify Answer:

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

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Example 2-5b Simplify Answer:

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

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

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

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

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

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End of Lesson 2

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

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

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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

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

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

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

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Example 3-2a Find Factor. The GCF is

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Example 3-2a Multiply. Simplify.Answer:

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

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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?

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

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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

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End of Lesson 3

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

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

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Example 4-1b Find Answer:

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

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

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Example 4-2c Find Answer:

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

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

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Example 4-3c Find Answer:

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

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

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

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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.

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

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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

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End of Lesson 4

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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

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

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Example 5-1b Find Answer:

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

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Example 5-2b Find Answer:

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

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Example 5-3b Find Answer:

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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.

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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.

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

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

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

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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:

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Example 5-5c Find Answer:The quotient is

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End of Lesson 5

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

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

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Example 6-1b Find Answer:

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

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Example 6-2b Answer: 5 Find

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

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

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

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

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

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Example 6-4b Simplify. Answer:

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Example 6-4c FindAnswer:

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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.

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

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Example 6-5c Find Answer:

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End of Lesson 6

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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

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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:

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Example 7-1b Find the LCM of Answer:

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

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Example 7-2b Find the LCM of Answer:

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

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

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Example 7-3c Find Answer:

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

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

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Example 7-4c Find Answer:

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

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Example 7-5b Multiply. Simplify.Answer:

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Example 7-5c Find Answer:

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

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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

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

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Example 7-6d Answer: C

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

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End of Lesson 7

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

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

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Example 8-1b Simplify Answer:

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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.

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

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

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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?

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

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

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Example 8-3c Answer: Simplify

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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.

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

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Example 8-4c Simplify Answer:

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End of Lesson 8

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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

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

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Example 9-1b Answer: –3 Solve

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

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

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Example 9-2c Answer: 8 Solve

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

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

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

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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.

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Example 9-3e Solve Answer: 4, –1

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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?

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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.

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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

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

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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.

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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

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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?

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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.

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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.

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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.

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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.

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

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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.

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Example 9-6c Solve Answer:no solutions

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

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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.

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Example 9-7c Solve Answer: –3

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End of Lesson 9

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