Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8-1
NAME ______________________________________________ DATE______________ PERIOD _____
8-1
Skills Practice
Practice
Multiplying and Dividing Rational Expressions
Multiplying and Dividing Rational Expressions
Simplify each expression. 21x3y 14x y
3x 2y
Simplify each expression.
b 5a
5ab3 25a b
1 9a2b3 27a b c 3a bc
1. ᎏ 2 2
2. ᎏ 2 2
(x6)3 3. ᎏ (x3)4
8y2(y6)3 2 4 4. ᎏ 4y24 y
x6
x2 x2 ⫺ 4 (x ⫺ 2)(x ⫹ 1) x 1
9 18 2x ⫺ 6 x 3
6. ᎏᎏ
5. ᎏᎏ 2
x4 ⫹ x3 ⫺ 2x2 x 2 x ⫺x x
7. ᎏ 2 2 5 ⭈ ᎏ 5
a⫹y 6
2 4 y⫹a 3
9. ᎏ ⭈ ᎏ 8 2
10(ef)3 8e f
1 3g y
3x2 x⫹2
3x x ⫺4
x(x 2) 13. ᎏ ⫼ ᎏ 2
w2 ⫺ 5w ⫺ 24 w⫹1
w2 ⫺ 6w ⫺ 7 w⫹3
15. ᎏᎏ ⭈ ᎏᎏ
a⫺y w⫹n
w2 ⫺ n2 y⫺a
5s2 s ⫺4
Glencoe Algebra 2
19.
c2 ᎏᎏ 2d2 ᎏ c6 ⫺ᎏᎏ 5d
Chapter 8
5 2c d
4
x2 ⫺ 5x ⫺ 24 6x ⫹ 2x
80y4 49z v
25y5 14z v
32z 7 35v y
q2 ⫹ 2q 6q
2
q q2 ⫺ 4 3q 2(q 2)
14. ᎏ ⫼ ᎏ 2
t2 ⫹ 19t ⫹ 84 4t ⫺ 4
t 12 2t ⫺ 2 t ⫹ 9t ⫹ 14 2(t 2)
16. ᎏᎏ ⭈ ᎏᎏ 2
(4a 5)(a 4) 4a ⫹ 5 16a2 ⫹ 40a ⫹ 25 18. ᎏᎏᎏ ⫼ ᎏᎏ 3a2 ⫺ 10a ⫺ 8 a2 ⫺ 8a ⫹ 16 3a 2
20.
a2 ⫺ b2 ᎏᎏ 4a ᎏ a⫹b ᎏᎏ 2a
8
5x2 8⫺x
5x 2
5x 1 25x2 ⫺ 1 x ⫺ 10x ⫹ 25 2(x 5)
13. ᎏ 7 ⫼ ᎏ 5 2 2
冢w 冣
3 24x2 xy w 3w
15. ᎏ ⫼ ᎏ
2 6x2 ⫹ 12x 4x ⫹ 12 x(x 3)
17. ᎏᎏ ⫼ ᎏᎏ 2
14. ᎏ 2
12. ᎏ 5 7 ⫼ ᎏ 12 5 2
n2 ⫺ 6n 1 n n
x⫺5 10x ⫺ 2 2xy 3
1 s⫹2 10s 2s (s 2)
5ux 2 21yz
25x3 14u y
11. ᎏᎏ ⭈ ᎏ 2
12. ᎏ ⭈ ᎏᎏ 2
10. ᎏ ⭈ᎏ 2 5 3
(w 8)(w 7)
1 x2 ⫺ 5x ⫹ 4 17. ᎏᎏ ⫼ (3x2 ⫺ 3x) 2x ⫺ 8 6x
4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7g y
3 11. ᎏ 2 ⫼ 21g 2 2
6
⫺2u3y 15xz
3x ⫹ 6 x ⫺9
x⫹y 6
⫼ᎏ 5
16. ᎏ ⫼ ᎏᎏ 2
9 ⫺ a2 a ⫹ 5a ⫹ 6
2a ⫺ 6 5a ⫹ 10
a5y3 wy
5 2
18. ᎏᎏ ⫼ ᎏ 2
x2 ⫺ 9 ᎏᎏ 4
20. ᎏ 2(x 3) 3⫺x ᎏᎏ 8
a3w2 a 2w 2 w y y
1 x2 ⫺ y2 3 2(x y)
2s 3 s2 ⫺ 10s ⫹ 25 s⫹4 (s 4)(s 5)
2s2 ⫺ 7s ⫺ 15 (s ⫹ 4)
2x ⫹ 1 ᎏᎏ x
2x 1 4x
19. ᎏ 4⫺x ᎏᎏ x
x3 ⫹ 23 ᎏ ᎏ x2 ⫺ 2x (x ⫹ 2) ᎏ ᎏ x2 ⫹ 4x ⫹ 4
x 2 2x 4 x(x 2)
21. ᎏᎏ 3
22. GEOMETRY A right triangle with an area of x2 ⫺ 4 square units has a leg that measures 2x ⫹ 4 units. Determine the length of the other leg of the triangle.
x 2 units
x2 ⫹ 3x ⫺ 10 2x
23. GEOMETRY A rectangular pyramid has a base area of ᎏᎏ square centimeters x2 ⫺ 3x x ⫺ 5x ⫹ 6
and a height of ᎏᎏ centimeters. Write a rational expression to describe the 2
x5 volume of the rectangular pyramid. cm3
ab 2
6
Glencoe Algebra 2
Chapter 8
Answers
9
Glencoe Algebra 2
(Lesson 8-1)
A3
24e3 5f
6e 9. ᎏ 2 ⭈ ᎏ 5
3m 2n
8. ᎏ ⭈ ᎏ
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
7. ᎏᎏ 2
mn 2
v5 3v 2
n5 n⫺6
8. ᎏ ⭈ ᎏ 10. ᎏ ⭈ ᎏ n w
n3
25 ⫺ v2 3v ⫺ 13v ⫺ 10
Answers
⫺ 24a a 8 3a ⫹ 12a a 4 3a2
3. ᎏᎏ 2
2k2 ⫺ k ⫺ 15 2k 5 k ⫺9 k3
4. ᎏᎏ 2 6. ᎏᎏ 4 3
5. ᎏ
10y2 ⫹ 15y 2y 3 35y ⫺ 5y 7y 1
4m 4n 2 9
(2m3n2)3 ⫺18m n
2. ᎏᎏ 5 4
1. ᎏ 4 4 2
Lesson 8-1
Chapter 8
NAME ______________________________________________ DATE______________ PERIOD _____
8-2
Study Guide and Intervention
NAME ______________________________________________ DATE______________ PERIOD _____
8-2
(continued)
Skills Practice
Adding and Subtracting Rational Expressions Add and Subtract Rational Expressions
Adding and Subtracting Rational Expressions Find the LCM of each set of polynomials.
To add or subtract rational expressions,
follow these steps. Step Step Step Step Step
1 2 3 4 5
If necessary, find equivalent fractions that have the same denominator. Add or subtract the numerators. Combine any like terms in the numerator. Factor if possible. Simplify if possible.
6 2x 2x 12
Example
2 x 4
Simplify . 2 2
6 2 ᎏᎏ ⫺ᎏ 2x2 ⫹ 2x ⫺ 12 x2 ⫺ 4 2 (x ⫺ 2)(x ⫹ 2)
6x ⫹ 12 ⫺ 4x ⫺ 12 2(x ⫹ 3)(x ⫺ 2)(x ⫹ 2)
Distributive Property
2x 2(x ⫹ 3)(x ⫺ 2)(x ⫹ 2)
Combine like terms.
x (x ⫹ 3)(x ⫺ 2)(x ⫹ 2)
Simplify.
A6
⫽ ᎏᎏᎏ
12 5y
15b 4a2 9b2 5ac 3abc
3. ᎏ ⫺ ᎏ
Chapter 8
2 x⫺3
x1 1 x ⫺ 1 (x 1)(x 3)
3 x⫹2
4x ⫹ 5 4x 14 3x ⫹ 6 3x ⫹ 6
2. ᎏ ⫺ ᎏ
4. ᎏ ⫹ ᎏ
4 4x ⫺ 4x ⫹ 1
2x 2 9x 4 5x 20x ⫺ 5 (2x 1)(2x 1)
6. ᎏᎏ ⫺ ᎏᎏ 2 2 2
14
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
4 x⫺1 x ⫺1 x1
5. ᎏᎏ ⫹ᎏ 2 2
2 12z 2y 5yz 5y z
11. ᎏ2 ⫺ ᎏ 2
Exercises
y 3
2c 5 3
2c ⫺ 7 3
Simplify each expression. 1. ᎏ ⫹ ᎏ
3 8p q
13 5 4p q 8p q
2 m n
5 2 5m 2 n m n
7 4gh
3 4h
8. ᎏ 2 2 ⫹ ᎏ 2
10. ᎏ ⫹ ᎏ 2 2
7h 3g 4gh
12. ᎏ ⫹ ᎏ 2 2 ᎏᎏ
2 a⫹2
a6 3 2a 2a(a 2)
14. ᎏ ⫺ ᎏ
3 w⫺3
3w 7 2 w ⫺ 9 (w 3)(w 3)
16. ᎏ ⫹ ᎏ
m m⫺n
2m m n⫺m mn
18. ᎏ ⫹ ᎏ
13. ᎏ ⫺ ᎏ
15. ᎏ ⫺ ᎏ 2
17. ᎏ ⫺ ᎏ
1 x ⫹ 2x ⫹ 1
x2 x 1 x x⫹1 (x 1)
19. ᎏᎏ ⫹ ᎏ 2 2
n n⫺3
2n ⫹ 2 n ⫺ 2n ⫺ 3
21. ᎏ ⫹ ᎏᎏ 2
5 3b ⫹ d
3t 2⫺x
5 3t 5 x⫺2 x2
4z z⫺4
z ⫹ 4 5z 2 4z 16 z ⫹ 1 (z 4)(z 1)
2x ⫹ 1 x⫺5
2x 2 5x 2 4 x ⫺ 3x ⫺ 10 (x 5)(x 2)
20. ᎏ ⫺ ᎏᎏ 2
3 y ⫹ y ⫺ 12
2 y ⫹ 6y ⫹ 8
22. ᎏᎏ ⫺ ᎏᎏ 2 2
n2 n3 Chapter 8
2 15bd 6b 2d 3bd 3bd(3b d )
y 12 (y 4)(y 3)(y 2)
15
Glencoe Algebra 2
(Lesson 8-2)
⫽ ᎏᎏᎏ ⫽ ᎏᎏᎏ
5 5x 3y y xy
9. ᎏ ⫹ 4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Subtract the numerators.
3x ⫹ 3 x ⫹ 2x ⫹ 1
6. x2 ⫺ 3x ⫺ 4, x ⫹ 1 (x 4)(x 1)
The LCD is 2(x ⫹ 3)(x ⫺ 2)(x ⫹ 2).
6(x ⫹ 2) ⫺ 4(x ⫹ 3) 2(x ⫹ 3)(x ⫺ 2)(x ⫹ 2)
4a 3bc
5. t2 ⫺ 25, t ⫹ 5 (t 5)(t 5)
3 x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2 ⭈ 2(x ⫹ 3) 2(x ⫹ 3)(x ⫺ 2)(x ⫹ 2)
⫽ ᎏᎏᎏ
4y2 2y
4. 5a, a ⫺ 1 5a(a 1)
7. ᎏ ⫹ ᎏ
Factor the denominators.
⫽ ᎏᎏᎏ ⫺ ᎏᎏᎏ
⫺7xy 3x
3. 2x ⫺ 6, x ⫺ 3 2(x 3)
Simplify each expression.
⫽ ᎏᎏ ⫺ ᎏᎏ 6(x ⫹ 2) 2(x ⫹ 3)(x ⫺ 2)(x ⫹ 2)
2. 18a3bc2, 24b2c2 72a 3b 2c 2
Answers
6 2(x ⫹ 3)(x ⫺ 2)
1. 12c, 6c2d 12c 2d
Lesson 8-2
Chapter 8
NAME ______________________________________________ DATE______________ PERIOD _____
8-6
NAME ______________________________________________ DATE______________ PERIOD _____
8-6
Lesson Reading Guide
Study Guide and Intervention
Solving Rational Equations and Inequalities
Solving Rational Equations and Inequalities
Get Ready for the Lesson
Solve Rational Equations
A rational equation contains one or more rational expressions. To solve a rational equation, first multiply each side by the least common denominator of all of the denominators. Be sure to exclude any solution that would produce a denominator of zero.
Read the introduction to Lesson 8-6 in your textbook. • If you increase the number of songs that you download, will your total bill increase or decrease? increase
2 5
冢
3
6
2
0
and
. Check
3 6 0 z .
Step 2 The related equation is z 2
z (z 2)
.
1
2
3
4
5
6
3 z⫹2
6 z
Consider the following values of ᎏ ⫺ ᎏ for various test values of z. 3 z⫹2 3 If z ⫽ ⫺1, ᎏ ⫺ z⫹2
6 3 z⫹2 z 3 6 If z ⫽ 1, ᎏ ⫺ ᎏ ⫽ ⫺5. z⫹2 z
6 z 6 ᎏ ⫽ 9. z
If z ⫽ ⫺5, ᎏ ⫺ ᎏ ⫽ 0.2.
If z ⫽ ⫺3, ᎏ ⫺ ᎏ ⫽ ⫺1.
Using this information and your number line, write the solution of the inequality.
z 4 or 2 z 0
Remember What You Learned 3. How are the processes of adding rational expressions with different denominators and of solving rational expressions alike, and how are they different? Sample answer: They
42
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Algebra 2
are alike because both use the LCD of all the rational expressions in the problem. They are different because in an addition problem, the LCD remains after the fractions are added, while in solving a rational equation, the LCD is eliminated.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
0
4(x ⫹ 1) 4x ⫹ 4 ⫺25 ⫺5
冢 冣
Multiply each side by 10(x ⫹ 1). Multiply. Distributive Property Subtract 4x and 29 from each side. Divide each side by 5.
2 Original equation 5 2 ⱨ ᎏ x ⫽ ⫺5 5 2 ⱨ ᎏ Simplify. 5 2 ⫽ᎏ 5
⫽ᎏ
Exercises Solve each equation. 2y 3
y⫹3 6
4t ⫺ 3 5
1. ᎏ ⫺ ᎏ ⫽ 2 5 3m ⫹ 2 5m
2m ⫺ 1 2m
4 ⫺ 2t 3
2. ᎏ ⫺ ᎏ ⫽ 1 2
1 24
4 x⫺1
x⫹1 12
4. ᎏ ⫹ ᎏ ⫽ 4 5. ᎏ ⫽ ᎏ 7
2x ⫹ 1 3
x⫺5 4
13 5
1 2
3. ᎏ ⫺ ᎏ ⫽ ᎏ x x⫺2
8 3
4 x⫺2
6. ᎏ ⫹ ᎏ ⫽ 10
7. NAVIGATION The current in a river is 6 miles per hour. In her motorboat Marissa can travel 12 miles upstream or 16 miles downstream in the same amount of time. What is the speed of her motorboat in still water? Is this a reasonable answer? Explain.
42 mph; Sample answer: The answer is reasonable. The boat will travel 1 48 mph one way and 36 mph the other way. Therefore it will take of 3 an hour to travel 16 miles and 12 miles, respectively. 8. WORK Adam, Bethany, and Carlos own a painting company. To paint a particular house 1 2
alone, Adam estimates that it would take him 4 days, Bethany estimates 5 ᎏ days, and Carlos 6 days. If these estimates are accurate, how long should it take the three of them
2 3
to paint the house if they work together? Is this a reasonable answer? about 1
days; Sample answer: It is a reasonable answer. It will take each person 1
about 5 days to paint the house alone, so it should take about of the 3 time to paint the house together. Chapter 8
43
Glencoe Algebra 2
(Lesson 8-6)
A20
4 Step 3 Divide a number line into regions using the excluded values and the solution of the related equation. Draw dashed vertical lines on the number line below to show these regions.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
To solve this equation, multiply both sides by the LCD, which is Solving this equation will show that the only solution is ⫺4.
9 2 ᎏ⫹ᎏ 10 x⫹1 9 2 ᎏ⫹ᎏ 10 ⫺5 ⫹ 1 18 10 ᎏ⫺ᎏ 20 20 2 ᎏ 5
⫽ ⫽ ⫽ ⫽
Answers
9(x ⫹ 1) ⫹ 2(10) 9x ⫹ 9 ⫹ 20 5x x
2. Suppose that on a quiz you are asked to solve the rational inequality ᎏ ⫺ ᎏ ⬎ 0. z⫹2 z Complete the steps of the solution. Step 1 The excluded values are
冣
Original equation
9 2 2 10(x ⫹ 1) ᎏ ⫹ ᎏ ⫽ 10(x ⫹ 1) ᎏ 10 x⫹1 5
0
1. When solving a rational equation, any possible solution that results in in the denominator must be excluded from the list of solutions.
Chapter 8
2 x1
Solve .
2 9 2 ᎏ⫹ᎏ ⫽ᎏ 10 x⫹1 5
Read the Lesson
6 5 4 3 2 1
9 10
Example
• Will your actual cost per song increase or decrease? decrease
Lesson 8-6
Chapter 8
NAME ______________________________________________ DATE______________ PERIOD _____
