Download Study Guide and Review

Study Guide and Review
Study Guide
KeyConcepts
KeyVocabulary
Inverse Variation (Lesson 11-1)
x1
y2
_
• You can use _
x 2 = y 1 to solve problems involving inverse
variation.
asymptote (p. 679)
mixed expression (p. 714)
complex fraction (p. 714)
product rule (p. 671)
excluded value (p. 678)
rate problems (p. 723)
Rational Functions (Lesson 11-2)
• Excluded values are values of a variable that result
in a denominator of zero.
• If vertical asymptotes occur, it will be at excluded values.
extraneous solution (p. 721)
rational equation (p. 720)
inverse variation (p. 670)
rational expression (p. 684)
least common denominator
(LCD) (p. 708)
rational function (p. 678)
Rational Expressions (Lessons 11-3 and 11-4)
• Multiplying rational expressions is similar to multiplying
rational numbers.
least common multiple
(LCM) (p. 707)
• Divide rational expressions by multiplying by the reciprocal
of the divisor.
Dividing Polynomials (Lesson 11-5)
• To divide a polynomial by a monomial, divide each term of
the polynomial by the monomial.
Adding and Subtracting Rational Expressions
(Lesson 11-6)
• Rewrite rational expressions with unlike denominators using
the least common denominator (LCD). Then add or subtract.
Complex Fractions (Lesson 11-7)
• Simplify complex fractions by writing them as division
problems.
Solving Rational Equations (Lesson 11-8)
• Use cross products to solve rational equations with a single
fraction on each side of the equals sign.
VocabularyCheck
State whether each sentence is true or false. If false, replace
the underlined word, phrase, expression, or number to make
a true sentence.
2
1. The least common multiple for x - 25 and x - 5
is x - 5.
2. If the product of two variables is a nonzero constant, the
relationship is an inverse variation.
3. If the line x = a is a vertical asymptote of a rational
function, then a is an excluded value.
4. A rational expression is a fraction in which the numerator
and denominator are fractions.
x
are -2 and -3.
5. The excluded values for _
2
x + 5x + 6
3x
6
=_
has an extraneous solution, 2.
6. The equation _
x-2
x-2
7. A rational expression has one or more fractions in the
numerator and denominator.
StudyOrganizer
Be sure the Key Concepts
are noted in your Foldable.
work problems (p. 722)
_1
Chapter 11
Rational Functions
and Equations
2
2.
can be simplified to _
8. The expression _
3
_
3
4
9. A direct variation can be represented by an equation of the
form k = xy, where k is a nonzero constant.
2
10. The rational function y = _
+ 3 has a horizontal
asymptote at y = 3.
x-1
connectED.mcgraw-hill.com
727
Study Guide and Review Continued
Lesson-by-Lesson Review
11-11Inverse Variation
A.8
(pp. 670–676)
Solve. Assume that y varies inversely as x.
Example 1
11. If y = 4 when x = 1, find x when y = 12
If y varies inversely as x and y = 28 when x = 42,
find y when x = 56.
12. If y = -1 when x = -3, find y when x = -9
13. If y = 1.5 when x = 6, find y when x = -16
14. PHYSICS A 135-pound person sits 5 feet from the
center of a seesaw. How far from the center should
a 108-pound person sit to balance the seesaw?
Let x 1 = 42, x 2 = 56, and y 1 = 28. Solve for y 2.
y
x1 _
_
= 2
Proportion for inverse variation
y1
y
42 _
_
= 2
56
28
x2
Substitution
1176 = 56y 2
Cross multiply.
21 = y 2
Thus, y = 21 when x = 56.
11-22 Rational Functions
Preparation for AII/T.6
(pp. 678–683)
State the excluded value for each function.
Example 2
1
15. y = _
State the excluded value for the function y =
x-3
3
17. y = _
3x - 6
2
16. y = _
2x - 5
-1
18. y = _
2x + 8
4x + 16 = 0
4x + 16 - 16 = 0 - 16
38
, where x is the number of people in the study
y=_
4x = -16
x
group. Graph the function and describe the asymptotes.
2
x+4
21. __
2
x 2 + 10x + 21
22. __
3
2
y - 25
23. __
2
16xyz
x + x - 42x
3x 3
24. _
3
2
3x + 6x
x + 12x + 32
2
y + 3y - 10
2
4y
25. _
4
3
8y + 16y
State the excluded values for each function.
x
26. y = _
2
x + 9x + 18
10
27. y = _
2
6x + 7x - 3
728 | Chapter 11 | Study Guide and Review
x = -4
Subtract 16 from each side.
Simplify.
Divide each side by 4.
A.1
(pp. 684–690)
Simplify each expression.
2xy
20. _
4x + 16
Set the denominator equal to zero.
19. PIZZA PARTY Katelyn ordered pizza and soda for her
study group for $38. The cost per person y is given by
11-33 Simplifying Rational Expressions
1
_
.
Example 3
Simplify
a - 7a + 12
__
.
2
a 2 - 13a + 36
Factor and simplify.
(a - 3)(a - 4)
a 2 - 7a + 12
__
= __
a 2 - 13a + 36
(a - 9)(a - 4)
a-3
_
=
a-9
Factor.
Simplify.
11-44 Multiplying and Dividing Rational Expressions
Preparation for AII/T.1.b
(pp. 692–698)
Find each product or quotient.
Example 4
6x 2y 4 3x 3y 2
28. _ · _
2
2
Find 7b · 6a .
_ _
12
xy
x+3
3x - 6 _
29. _
· 2
2
x - 9 x - 2x
2
3x
_
30. x ÷ _
x+4
x 2 - 16
3b - 12
31. _
÷ (b 2 - 6b + 8)
b+4
9
b
2
6a
a 2b 2
_· _
_
= 42
9
b
9b
14a 2b
=_
3
7b 2
Simplify.
Example 5
_ _
2
x+5
Find x 2- 25 ÷
.
2a 2 + 7a - 15
9a 2 - 4
32. __ ÷ _
a+5
Multiply.
x -9
3a + 2
x-3
(x + 5)(x - 5)
x+5
x+5
x 2 - 25
_
÷_=_÷_
33. GEOMETRY Find the area
of the rectangle shown.Write
the answer in simplest form.
x2 - 9
2x
y
x-3
x-3
(x + 3)(x - 3)
2
1
1
(x + 5) (x - 5) x - 3
= __ · _
(x + 3)(x - 3)
x-5
=_
2
y
2x
11-55 Dividing Polynomials
x+5
1
1
Factor.
Multiply
by the
reciprocal.
Simplify.
x+3
A.2.b
(pp. 700–705)
Find each quotient.
Example 6
34. (x 3 - 2x 2 - 22x + 21) ÷ (x - 3)
Find (4x 2 + 17x - 1) ÷ (4x + 1).
x+4
35. (x 3 + 7x 2 + 10x - 6) ÷ (x + 3)
4x + 1 !"""""""""""""""""""""""""""""""""""""""""""""""""""""""
4x 2 + 17x - 1
36. (5x 2y 2 - 10x 2y + 5xy) ÷ 5xy
4_______
x2 + x
16x - 1
16x + 4
_______
-5
37. (48y + 8y + 7) ÷ (12y - 1)
2
2
38. GEOMETRY The area of a rectangle is x + 7x + 13. If
the length is (x + 4), what is the width of the rectangle?
Multiply x and 4x + 1.
Subtract, bring down -1.
Multiply 4 and 4x + 1.
Subtract.
5
The quotient is x + 4 - _
.
4x + 1
11-66 Adding and Subtracting Rational Expressions
Find each sum or difference.
5a
2a
-_
39. _
b
-3
2n
40. _
+_
2n - 3
2n - 3
3
1
42. _
+_
x+1
x-2
b
y
3
41. _
-_
y+1
y-3
43. DESIGN Miguel is decorating a model of a room that
8
2x
is _
feet long and _
feet wide. What is the
x+4
x+4
Preparation for AII/T.1.b
(pp. 706–713)
Example 7
_ _
2
2x + 1
Find x +
.
x+1
x+1
2x + 1
x 2 + 2x + 1
x
_
+_= _
x+1
x+1
x+1
(x + 1)(x + 1)
_
=
x+1
2
=x+1
Add the numerators.
Factor.
Simplify.
perimeter of the room?
connectED.mcgraw-hill.com
729
Study Guide and Review Continued
11-77 Mixed Expressions and Complex Fractions
Simplify each expression.
a 2b 4
_
c
44. _
3
a b
_
35
x-_
x+2
45. _
42
x+_
x 2 - 25
_
x-5
_
Example 8
x+3
_
x - 2x - 15
_
x
6
Simplify _
.
2
x +13
c2
x+2
46. _
Preparation for AII/T.1.b
(pp. 714–719)
47.
x2 - 4
6
y+9-_
Write as a division expression.
2
y+4+_
x+3
x 2 - 2x - 15
6
_
= _ ÷ __
y+4
__
y+1
x+3
_
x
x
6
x 2 - 2x - 15
__
x
= _ · __
2
x+3
6
48. FABRICS Donna makes tablecloths to sell at craft fairs.
A small one takes one-half yard of fabric, a medium one
takes five-eighths yard, and a large one takes one and
one-quarter yard.
x - 2x - 15
x
+
3
x
= _ · __
6
(x + 3)(x - 5)
x
=_
6(x - 5)
a. How many yards of fabric does she need to make a
tablecloth of each size?
b. One bolt of fabric contains 30 yards of fabric. Can
she use the entire bolt of fabric by making an equal
number of each type of tablecloth? Explain.
11-88 Rational Equations
Preparation for AII/T.4.c
(pp. 720–726)
Solve each equation. State any extraneous solutions.
Example 9
n+1
5n
1
+_
=_
49. _
Solve
50.
51.
52.
53.
6
n-2
3(n - 2)
4x
7 _
_
+_
= 7x - 14
3
2
12
11 _
1
_
+ 2 =_
2x
4x
4
1
1
2
_
-_
=_
x+4
x-1
x 2 + 3x - 4
n
1
_
=_
n-2
8
54. PAINTING Anne can paint a room in 6 hours. Oljay can
paint a room in 4 hours. How long will it take them to
paint the room working together?
x+2
3
1
_
+_=_
.
x 2 + 3x
x+3
x
x+2
3
1
_
+_ = _
x
x+3
x+2
3
1
x (x + 3) _
+ x(x + 3) _ = x(x + 3) _
x
x+3
x(x + 3)
(
)
x 2 + 3x
(
)
()
3 + x(x + 2) = 1(x + 3)
3 + x 2 + 2x = x + 3
x2 + x = 0
x(x + 1) = 0
x = 0 or x = -1
The solution is -1, and there is an extraneous solution of 0.
730 | Chapter 11 | Study Guide and Review