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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 Welcome to Interactive Chalkboard

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Splash Screen

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**Lesson 12-1 Inverse Variation Lesson 12-2 Rational Expressions **

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

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**Example 1 Graph an Inverse Variation **

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

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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 equation can be used to represent the people building a computer. Complete the table and draw a graph of the relation. p 2 4 6 8 10 12 t Example 1-1a

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**Solve the equation for the other values of p.**

Solve for Original equation Replace p with 2. Divide each side by 2. Simplify. Answer: p 2 4 6 8 10 12 t 6 3 2 1.5 1.2 1 Solve the equation for the other values of p. Example 1-1b

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**Graph the ordered pairs: (2, 6), (4, 3), (6, 2), (8, 1. 5), (10, 1**

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

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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 equation can be used to represent the people preparing the packages. Complete the table and draw a graph of the relation. Answer: p 2 4 6 8 10 12 t 18 9 4.5 3.6 3 p 2 4 6 8 10 12 t Example 1-1d

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Answer: Example 1-1e

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**Graph an inverse variation in which y varies inversely as x and**

Solve for k. Inverse variation equation The constant of variation is 4. Example 1-2a

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**Choose values for x and y whose product is 4. x y –4 –1 –2**

undefined 1 4 2 Example 1-2b

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**Graph an inverse variation in which y varies inversely as x and**

Answer: Example 1-2c

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**If y varies inversely as x and find x when**

Method 1 Use the product rule. Product rule for inverse variations Divide each side by 15. Simplify. Example 1-3a

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**Method 2 Use a proportion.**

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

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**If y varies inversely as x and find x when**

Answer: 8 Example 1-3c

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**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: Example 1-4a

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**If y varies inversely as x and find y when**

Answer: –25 Example 1-4b

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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? Example 1-5a

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**Answer: The 2-kilogram weight should be 9.6 meters from the fulcrum.**

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

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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? Answer: 1 m Example 1-5c

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End of Lesson 1

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**Example 1 One Excluded Value Example 2 Multiple Excluded Values **

Example 3 Use Rational Expressions Example 4 Expression Involving Monomials Example 5 Expression Involving Polynomials Example 6 Excluded Values Lesson 2 Contents

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**State the excluded value of**

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

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**State the excluded value of**

Answer: –3 Example 2-1b

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**State the excluded values of**

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. Example 2-2a

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**State the excluded values of**

Answer: 2, 3 Example 2-2b

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**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. The original mechanical advantage was 5. Example 2-3a

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Use the expression for mechanical advantage to write an expression for the mechanical advantage in the new situation. 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. Example 2-3b

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

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**a. Use the formula to find the mechanical advantage.**

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 formula to 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 Example 2-3d

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**The GCF of the numerator and denominator is**

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

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Simplify Answer: Example 2-4b

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**Divide the numerator and denominator by the GCF, x – 7.**

Simplify Factor. Divide the numerator and denominator by the GCF, x – 7. 1 Answer: Simplify Example 2-5a

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Simplify Answer: Example 2-5b

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**Simplify State the excluded values of x.**

Factor. Divide the numerator and denominator by the GFC, x + 4. 1 Simplify. Answer: Example 2-6a

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**Exclude the values for which equals 0.**

The denominator cannot equal zero. Factor. Zero Product Property Example 2-6b

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**Check. Verify the excluded values by substituting**

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

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**Answer: The expression is undefined when and Therefore,**

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

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**Simplify State the excluded values of w.**

Answer: Example 2-6e

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End of Lesson 2

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**Example 1 Expressions Involving Monomials **

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

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**Method 1 Divide by the greatest common factor after multiplying.**

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

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**Method 2 Divide the common factors before multiplying.**

Divide by common factors and z. 1 x 6 z 2 y 3 Multiply. Answer: Example 3-1a

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**Divide by common factors and r.**

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

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

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**Find Factor the numerator. The GCF is Simplify. Answer: x 2 1**

Example 3-2a

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Find Factor. The GCF is 1 Example 3-2a

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Multiply. Simplify. Answer: Example 3-2a

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

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Space The velocity that a spacecraft must have in order to escape Earth’s 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? Example 3-3a

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**Answer: The escape velocity is 11,200 meters per second.**

1120 10 1 Simplify. Multiply. Answer: The escape velocity is 11,200 meters per second. Example 3-3a

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**Answer: 1188 kilometers per hour**

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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**Example 1 Expression Involving Monomials **

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

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**Multiply by the reciprocal of**

Find Multiply by the reciprocal of Divide by common factors 5, 6, and x. 1 4 15 x3 Answer: Simplify. Example 4-1a

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Find Answer: Example 4-1b

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**Multiply by the reciprocal of**

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

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

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Find Answer: Example 4-2c

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**Multiply by the reciprocal of**

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

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

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Find Answer: Example 4-3c

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**Multiply by the reciprocal,**

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

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

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

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**Use the formula for rate, time, and distance.**

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. Example 4-5a

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**Answer: Thus, the speed of the aircraft was about 116 miles per hour.**

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

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**Answer: about 149 miles per hour**

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 Example 4-5c

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End of Lesson 4

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**Example 1 Divide a Binomial by a Monomial **

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

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**Write as a rational expression.**

Find Write as a rational expression. Divide each term by 2x. 2x 9 1 Simplify each term. Simplify. Answer: Example 5-1a

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Find Answer: Example 5-1b

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**Write as a rational expression.**

Find Write as a rational expression. Divide each term by 3y. 2y 1 3 Simplify each term. y Simplify. Answer: Example 5-2a

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Find Answer: Example 5-2b

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**Write as a rational expression.**

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

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Find Answer: Example 5-3b

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Find Step 1 Divide the first term of the dividend, x2, by the first term of the divisor, x. x Multiply x and x – 2. Subtract. Example 5-4a

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**Step 2. Divide the first term of the partial dividend,**

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

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**Answer:. The quotient of. is**

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

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**Answer: The quotient is with a remainder of 2.**

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

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**Rename the x2 term by using a coefficient of 0.**

Find Rename the x2 term by using a coefficient of 0. Example 5-5a

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**Answer: Multiply x2 and x – 5. Subtract and bring down –34x.**

Multiply –9 and x – 5. Answer: Example 5-5b

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**Answer: The quotient is**

Find Answer: The quotient is Example 5-5c

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End of Lesson 5

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**Example 1 Numbers in Denominator Example 2 Binomials in Denominator **

Example 3 Find a Perimeter Example 4 Subtract Rational Expressions Example 5 Inverse Denominators Lesson 6 Contents

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**The common denominator is 15.**

Find The common denominator is 15. Add the numerators. Divide by the common factor, 5. 4 3 Simplify. Answer: Example 6-1a

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Find Answer: Example 6-1b

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**The common denominator is c + 2.**

Find The common denominator is c + 2. Factor the numerator. Divide by the common factor, c + 2. 1 Simplify. Answer: Example 6-2a

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Find Answer: 5 Example 6-2b

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**Geometry Find an expression for the perimeter of rectangle WXYZ.**

Perimeter formula Example 6-3a

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**The common denominator is**

Distributive Property Combine like terms. Factor. Example 6-3b

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**Answer: The perimeter can be represented by the expression**

Example 6-3c

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**18 Geometry Find an expression for the perimeter of rectangle PQRS.**

Answer: 18 Example 6-3d

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**The common denominator is**

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

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Simplify. Answer: Example 6-4b

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Find Answer: Example 6-4c

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**Rewrite using common denominators.**

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

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**The common denominator is**

Simplify. Answer: Example 6-5b

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Find Answer: Example 6-5c

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End of Lesson 6

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**Example 1 LCM of Monomials Example 2 LCM of Polynomials **

Example 3 Monomial Denominators Example 4 Polynomial Denominators Example 5 Binomials in Denominators Example 6 Polynomials in Denominators Lesson 7 Contents

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**Find the prime factors of each coefficient and variable expression.**

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: Example 7-1a

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Find the LCM of Answer: Example 7-1b

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**Express each polynomial in factored form.**

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

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Find the LCM of Answer: Example 7-2b

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**Factor each denominator and find the LCD.**

Since the denominator of is already 5z, only needs to be renamed. Example 7-3a

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**Distributive Property**

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

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Find Answer: Example 7-3c

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**Factor the denominators.**

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

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

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Find Answer: Example 7-4c

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

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Multiply. Simplify. Answer: Example 7-5b

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Find Answer: Example 7-5c

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**Multiple-Choice Test Item Find**

A B C D Example 7-6a

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**Step 1 Factor each denominator and find the LCD.**

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

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**Step 2. Change each rational expression into an**

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

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Answer: C Example 7-6d

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**Multiple-Choice Test Item Find**

A B C D Answer: C Example 7-6e

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End of Lesson 7

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**Example 1 Mixed Expression to Rational Expression **

Example 2 Complex Fraction Involving Numbers Example 3 Complex Fraction Involving Monomials Example 4 Complex Fraction Involving Polynomials Lesson 8 Contents

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**Distributive Property**

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

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Simplify Answer: Example 8-1b

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**Baking Suppose Katelyn bought 2 pounds of chocolate chip cookie dough**

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. Example 8-2a

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**Convert pounds to ounces and divide by common units.**

Simplify. Express each term as an improper fraction. Example 8-2b

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**Answer: Katelyn can make 21 cookies.**

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

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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? Answer: 27 cookies Example 8-2d

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**Rewrite as a division sentence.**

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

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**Divide by common factors a, b, and c2.**

1 Simplify. Answer: Example 8-3b

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Simplify Answer: Example 8-3c

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**The LCD of the fractions in the numerator is**

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

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**Rewrite as a division sentence.**

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

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Simplify Answer: Example 8-4c

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End of Lesson 8

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**Example 1 Use Cross Products Example 2 Use the LCD **

Example 3 Multiple Solutions Example 4 Work Problem Example 5 Rate Problem Example 6 No Solution Example 7 Extraneous Solution Lesson 9 Contents

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**Distributive Property**

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

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Solve Answer: –3 Example 9-1b

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

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**Subtract 1 from each side.**

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

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Solve Answer: 8 Example 9-2c

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

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

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**Check. Check solutions by substituting each value in**

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

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**Check. Check solutions by substituting each value in**

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

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Solve Answer: 4, –1 Example 9-3e

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TV Installation On Saturdays, Lee helps her father install satellite TV systems. The jobs normally take Lee’s 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? Example 9-4a

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**Explore. Since it takes Lee’s father**

Explore Since it takes Lee’s father hours 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 Lee’s work + her father’s work = 1 job. Example 9-4b

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**Plan. The time that both of them worked was hours**

Plan The time that both of them worked was hours. Each rate multiplied by this time results in the amount of work done by each person. Solve Lee’s her father’s total work plus work equals work. 1 Example 9-4c

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**Distributive Property**

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

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**Answer: The job would take Lee or hours by herself.**

Add –6t to each side. Divide each side by 4. Answer: The job would take Lee or hours by herself. Examine This seems reasonable because the combined efforts of the two took longer than half of her father’s usual time. Example 9-4e

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**Driveways Shawna earns extra money by shoveling driveways**

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 Example 9-4f

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**Transportation The schedule for the Washington, D. C**

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? Example 9-5a

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**Determine the rates of both trains. The total distance is 19.4 miles.**

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, miles. Example 9-5b

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**t min d = r t t r Train 1 Train 2 The sum of the distances is 19.4.**

The LCD is 432. Example 9-5c

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**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. Example 9-5d

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**Transportation Two cyclists are riding on a 5-mile circular bike trail**

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. Example 9-5e

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**Distributive Property**

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

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**Subtract 2 from each side.**

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. Example 9-6b

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Solve Answer: no solutions Example 9-6c

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**Distributive Property**

Solve Original equation The LCD is x – 2. 1 Distributive Property Example 9-7a

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**Subtract 4 from each side.**

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. Example 9-7b

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Solve Answer: –3 Example 9-7c

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End of Lesson 9

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Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION.

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