Answers
To multiply two rational expressions such as multiply the numerators and the denominators. A. 4. ... Answers (Anticipation Guide and Lesson 11-1) ...
Multiplying and Dividing Rational Expressions
May 4 2014 those skills to multiply and divide rational expressions. ... Your Turn. Find the quotients and any excluded values. 4. x + 11.
Skills Practice - Multiplying and Dividing .Rational Expressions
GEOMETRY The expressions 5x 20 and 10. 2' x + 4 '. ~x represent the lengths of the sides of a triangle. Write a simplified expression for the perimeter of the
Rational Expression Worksheet #1: Simplifying
11 ??? !! 4x+4 x²+4x+3. = 4(x+1). The box and ag. 2. Rational Expression Worksheet #2: Simplifying Add or subtract these rational expressions. 11.
Chapter 11 Resource Masters
Homework Practice Workbook Answers For Workbooks The answers for Chapter 11 of these workbooks can be ... Multiplying and Dividing Rational Expressions.
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11-4 Practice. 1. Find each product or quotient. 18x2 15y3 4. P b² + 5b + 4 b236. 28a2 21a3. 762. 356. Multiplying and Dividing Rational Expressions.
Chapter 11 Resource Masters
Homework Practice Workbook Answers For Workbooks The answers for Chapter 11 of these workbooks can be ... Multiplying and Dividing Rational Expressions.
There Must Be a Rational Explanation
Simplify the answer when possible. Carnegie Learning. Chapter 10 Skills Practice 615. 10. 11. ... Multiplying and Dividing Rational Expressions.
Chapter 9: Rational Expressions and Equations
Multiplying and Dividing Rational Expressions (pp. 472–478) 4–6. 11 12. 7. 13. 2. Practice and Apply indicates increased difficulty.
With Great Power . . .
Chapter 11 Skills Practice. 11. LESSON 11.1 Skills Practice page 4. 9. y 5 x2 1 3 Answers will vary but should be of the form f(x) 5 ?.
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Answers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11 A1 Glencoe Algebra 1
Chapter Resources
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAMEDATE PERIOD
Chapter 11 3 Glencoe Algebra 1Anticipation GuideRational Expressions and EquationsBefore you begin Chapter 11
• Read each statement. • Decide whether you Agree (A) or Disagree (D) with the statement. • Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).After you complete Chapter 11
• Reread each statement and complete the last column by entering an A or a D. • Did any of your opinions about the statements change from the first column? • For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.11Step 1STEP 1
A, D, or NSStatementSTEP 2
A or D
1. Since a direct variation can be written as y = kx, an inverse
variation can be written as y = x ? k . D2. A rational expression is an algebraic fraction that contains
a radical. D3. To multiply two rational expressions, such as 2xy2
3c and 3 c 2 5y multiply the numerators and the denominators. A4. When solving problems involving units of measure,
dimensional analysis is the process of determining the units of the final answer so that the units can be ignored while performing calculations. D5. To divide (4x2 + 12x) by 2x, divide 4x2 by 2x and 12x by 2x.
A6. To find the sum of 2a
(3a - 4) and 5 (3a - 4) , first add the numerators and then the denominators. D7. The least common denominator of two rational expressions
will be the least common multiple of the denominators. A8. A complex fraction contains a fraction in its numerator
or denominator. A9. The fraction
(a ? b ) ( c ? d ) can be rewritten as ac bd D10. Extraneous solutions are solutions that can be eliminated
because they are extremely high or low. DStep 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAMEDATE PERIOD
Lesson 11-1
Chapter 11 5 Glencoe Algebra 1Both methods show that x2 = 8 ? 3 when y = 18.Exercises
Determine whether each table or equation represents an inverse or a direct variation. Explain. 1. x y 3 6 5 10 8 1612 24 2. y = 6x 3. xy = 15
Assume that y varies inversely as x. Write an inverse variation equation that relates x and y. Then solve.4. If y = 10 when x = 5, 5. If y = 8 when x = -2,
find y when x = 2. xy = 50; 25 find y when x = 4. xy = -16; -46. If y = 100 when x = 120, 7. If y = -16 when x = 4,
find x when y = 20. xy = 12,000; 600 find x when y = 32. xy = -64;-28. If y = -7.5 when x = 25, find y when x = 5.
xy = -187.5; -37.59. DRIVING The Gerardi family can travel to Oshkosh, Wisconsin, from Chicago, Illinois,
in 4 hours if they drive an average of 45 miles per hour. How long would it take them if they increased their average speed to 50 miles per hour? 3.6 h10. GEOMETRY For a rectangle with given area, the width of the rectangle varies inversely
as the length. If the width of the rectangle is 40 meters when the length is 5 meters, find the width of the rectangle when the length is 20 meters. 10 mStudy Guide and Intervention
Inverse Variation
Identify and Use Inverse Variations An inverse variation is an equation in theform of y = k ? x or xy = k. If two points (x1, y1) and (x2, y2) are solutions of an inverse variation,
then x1 ∙ y1 = k and x2 ∙ y2 = k. Product Rule f or Inverse Variationx1 ∙ y1 = x2 ∙ y2 From the product rule, you can form the proportion x1 ? x2 = y1 ? y2 . If y varies inversely as x and y = 12 when x = 4, find x when y = 18.Method 1 Use the product rule.
x1 ∙ y1 = x2 ∙ y2 Product rule for inverse variation 4 ∙ 12 = x2 ∙ 18 x1 = 4, y1 = 12, y2 = 18 4818 = x
2 Divide each side by 18.
8 ? 3 = x
2 Simplify.Method 2 Use a proportion.
x 1 ? x 2 = y 2 ? y 1Proportion for inverse variation
4 ? x 2 = 18 12 x1 = 4, y1 = 12, y2 = 1848 = 18x2 Cross multiply.
8 ? 3 = x2 Simplify.11-1Example
direct variation; of the form y = kxdirect variation; of the form y = kxinverse variation; of the form xy = kAnswers (Anticipation Guide and Lesson 11-1)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11 A2 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAMEDATE PERIOD
Chapter 11 6 Glencoe Algebra 1Study Guide and Intervention (continued)Inverse Variation
Graph Inverse Variations Situations in which the values of y decrease as the values of x increase are examples of inverse variation. We say that y varies inversely as x, or y is inversely proportional to x.Suppose you drive
200 miles without stopping. The time
it takes to travel a distance varies inversely as the rate at which you travel. Let x = speed in miles per hour and y = time in hours. Graph the variation.The equation xy = 200 can be used to
represent the situation. Use various speeds to make a table. xy O 20 4060
30
20 10
Graph an inverse
variation in which y varies inversely as x and y = 3 when x = 12.Solve for k.
xy = k Inverse variation equation12(3) = k x = 12 and y = 3
36 = k Simplify.
Choose values for x and y, which have a
product of 36. xy O 12 2424
12
Exercises
Graph each variation if y varies inversely as x.
1. y = 9 when x = -3 2. y = 12 when x = 4 3. y = -25 when x = 5
xy O 2412 -12 -24 -12 -24 12 24 x y O3216 -16 -32 -16-32 16 32 xy O
50-50-100 100100
50-50 -100
4. y = 4 when x = 5 5. y = -18 when x = -9 6. y = 4.8 when x = 5.4
xy O 20 10 -10 -20 -10-20 10 20 xy O 3618 -18 -36 -18 -36 18 36 xy O 7.2 3.6 -3.6 -7.2-3.6-7.2 3.6 7.2
11-1Example 1
Example 2
x y 10 20 20 1030 6.7
40 550 4
60 3.3x y -6 -6 -3 -12 -2 -18
2 18 3 126 6Inverse Variation Equation
an equation of the form xy = k, where k ≠ 0 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAMEDATE PERIOD
Lesson 11-1
Chapter 11 7 Glencoe Algebra 1Skills PracticeInverse Variation Determine whether each table or equation represents an inverse or a direct variation. Explain. 1. x y 0.5 8 1 4 2 24 1 2. xy = 2
? 3 3. -2x + y = 0 Assume that y varies inversely as x. Write an inverse variation equation that relates x and y. Then graph the equation.4. y = 2 when x = 5 5. y = -6 when x = -6
6. y = -4 when x = -12 7. y = 15 when x = 3
Solve. Assume that y varies inversely as x.
8. If y = 4 when x = 8, 9. If y = -7 when x = 3,
find y when x = 2. xy = 32; 16 find y when x = -3. xy = -21; 710. If y = -6 when x = -2, 11. If y = -24 when x = -3,
find y when x = 4. xy = 12; 3 find x when y = -6. xy = 72; -1212. If y = 15 when x = 1, 13. If y = 48 when x = -4,
find x when y = -3. xy = 15; -5 find y when x = 6. xy = -192; -3214. If y = -4 when x = 1
? 2 , find x when y = 2. xy = -2; -1 11-1 xy O8-8-16 1616
8 -8 -16 xy O8-8-16
16168 -8 -16 xy O
10-10-20
202010
-10 -20 xy O 4-4-8 884 -4 -8 inverse, xy = 4inverse, xy = 2 ? 3 direct, y = 2x xy = 10xy = 36 xy = 48xy = 45
Answers (Lesson 11-1)
Answers
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.Chapter 11 A3 Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAMEDATE PERIOD
Lesson 11-1
Chapter 11 9 Glencoe Algebra 11.PHYSICAL SCIENCE The illumination I produced by a light source varies inversely as the square of the distance d from the source. The illumination produced 5 feet from the light source is80 foot-candles.
Id 2=k 80(5)2=k
2000=k
Find the illumination produced 8 feet
from the same source.31.25 foot-candles
2. MONEY A formula called the Rule of
72 approximates how fast money will
double in a savings account. It is based on the relation that the number of years it takes for money to double varies inversely as the annual interest rate.Use the information in the table to write
the Rule of 72 formula. yr = 723. ELECTRICITY The resistance, in ohms,
of a certain length of electric wire varies inversely as the square of the diameter of the wire. If a wire 0.04 centimeter in diameter has a resistance of 0.60 ohm, what is the resistance of a wire of the same length and material that is 0.08 centimeters in diameter?0.15 ohm
4.BUSINESS In the manufacturing of a
certain digital camera, the cost of producing the camera varies inversely as the number produced. If 15,000 cameras are produced, the cost is $80 per unit.Graph the relationship and label the
point that represents the cost per unit to produce 25,000 cameras. $485. SOUND The sound produced by a string
inside a piano depends on its length. The frequency of a vibrating string varies inversely as its length. a. Write an equation that represents the relationship between frequency f and length ?. Use k for the constant of variation.b. If you have two different length strings, which one vibrates more quickly (that is, which string has a greater frequency)?
The shorter string vibrates
more quickly than the longer string.c. Suppose a piano string 2 feet long vibrates 300 cycles per second. What would be the frequency of a string 4 feet long?
150 cycles per second
Word Problem Practice
Inverse Variation
11-1 Years to DoubleMoneyAnnual Interest
Rate (percent) 18 414.4 5
12 610.29 7
Price per Unit ($)200300100
0Units Produced
(thousands)10 20 30y x25,000,48
f × ℓ = k or f = k Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. NAMEDATE PERIOD
Chapter 11 8 Glencoe Algebra 1Practice Inverse Variation Determine whether each table or equation represents an inverse or a direct variation. Explain. 1. x y0.25 40
0.5 20
2 58 1.25 2.
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