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9
PRECALCULUS REVIEW
9.1
Exponents and Radicals
9.2
Algebraic Expressions
9.3
Algebraic Fractions
9.4
Inequalities and Absolute Value
This chapter contains a brief review of the algebra you will need
in the study of calculus. To solve many calculus problems, it is often
necessary to simplify certain algebraic expressions before moving
on to the next step. Thus it is important that you reestablish familiarity with this process. This chapter also contains a short review of
inequalities and absolute value; their uses range from describing
the domains of functions to formulating practical problems.
How much money is needed to purchase at least 100,000
shares of the Starr Communications Company? Corbyco,
a giant conglomerate, wishes to purchase a minimum of
100,000 shares of the company. In Example 4, page
564, you will see how Corbyco’s management determines
how much money they will need for the acquisition.
539
9.1
9 䡲 PRECALCULUS REVIEW
Exponents and Radicals
E XPONENTS
AND
R ADICALS
This section reviews the properties of exponents and radicals. Recall that if
b is any real number and n is a positive integer, then the expression b n (read
‘‘b to the power n’’) is defined as the number
bn ⫽ b ⴢ b ⴢ b ⴢ ⭈ ⭈ ⭈ ⴢ b
冦
540
n factors
The number b is called the base, and the superscript n is called the power of
the exponential expression b n. For example,
25 ⫽ 2 ⴢ 2 ⴢ 2 ⴢ 2 ⴢ 2 ⫽ 32
冉 冊 冉 冊冉 冊冉 冊
2
3
and
3
⫽
2
3
2
3
2
8
⫽
3
27
If b ⬆ 0, we define
b0 ⫽ 1
For example, 20 ⫽ 1 and (⫺앟)0 ⫽ 1, but the expression 00 is undefined.
Next, recall that if n is a positive integer, then the expression b1/n is defined
to be the number that, when raised to the nth power, is equal to b. Thus,
(b1/n)n ⫽ b
n
b.
Such a number, if it exists, is called the nth root of b, also written 兹
Observe that the nth root of a negative number is not defined when n is even.
For example, the square root of ⫺2 is not defined because there is no real
number b such that b2 ⫽ ⫺2. Also, given a number b, more than one number
might satisfy our definition of the nth root. For example, both 3 and ⫺3
squared equal 9, and each is a square root of 9. So, to avoid ambiguity, we
define b1/n to be the positive nth root of b whenever it exists. Thus, 兹9 ⫽
91/2 ⫽ 3.
Next, recall that if p/q ( p, q, positive integers with q ⬆ 0) is a rational
number in lowest terms, then the expression b p/q is defined as the number
q
(b1/q)p or, equivalently, 兹
b p, whenever it exists. Expressions involving negative
rational exponents are taken care of by the definition
b⫺p/q ⫽
1
b p/q
Examples are
23/2 ⫽ (21/2)3 앒 (1.4142)3 앒 2.8283
and
4⫺5/2 ⫽
1
1
1
1
⫽
⫽ ⫽
45/2 (41/2)5 25 32
The rules defining the exponential expression an, where a ⬎ 0 for all rational
values of n, are given in Table 9.1.
9.1 䡲 E X P O N E N T S A N D R A D I C A L S
541
Table 9.1 Definitions of Exponents
Definition of a n (a ⬎ 0)
Definition of a n (a ⬎ 0)
Example
Integer exponent: If n is a
positive integer, then
an ⫽ a ⴢ a ⴢ a ⴢ ⭈ ⭈ ⭈ ⴢ a
(n factors of a)
Fractional exponent:
a. If n is a positive integer,
then
25 ⫽ 2 ⴢ 2 ⴢ 2 ⴢ 2 ⴢ 2
(5 factors)
161/2 ⫽ 兹16
n
a1/n or 兹
a
⫽ 32
denotes the nth root of a.
Zero exponent: If n is equal
to zero, then
a0 ⫽ 1
(00 is not defined.)
Example
⫽4
b. If m and n are positive integers, then
70 ⫽ 1
82/3 ⫽ (兹8)2
3
n
a m/n ⫽ 兹a m ⫽ (兹
a)m
n
⫽4
Negative exponent: If n is a
positive integer, then
a⫺n ⫽
1
an
(a ⫽
/ 0)
c. If m and n are positive integers, then
6⫺2 ⫽
1
62
⫽
1
36
a⫺m/n ⫽
1
a m/n
(a ⫽
/ 0)
9⫺3/2 ⫽
1
93/2
⫽
1
27
The first three definitions in Table 9.1 are also valid for negative values
of a, whereas the fourth definition is valid only for negative values of a when
n is odd. Thus,
3
(⫺8)1/3 ⫽ 兹
⫺8 ⫽ ⫺2
1/2
(⫺8) has no real value
(n is odd.)
(n is even.)
Finally, note that it can be shown that an has meaning for all real numbers n.
For example, using a pocket calculator with a ‘‘y x ’’ key, we see that 2兹2苵 앒
2.665144.
The five laws of exponents are listed in Table 9.2.
Table 9.2 Laws of Exponents
Law
Example
1. a m ⴢ a n ⫽ a m⫹n
2.
x 2 ⴢ x 3 ⫽ x 2⫹3 ⫽ x 5
am
⫽ a m⫺n (a ⫽
/ 0)
an
3. (a m ) n ⫽ a mⴢn
4. (ab) ⫽ a ⴢ b
n
5.
冉冊
a
b
n
n
⫽
x7
⫽ x 7⫺4 ⫽ x 3
x4
(x 4)3 ⫽ x 4ⴢ3 ⫽ x 12
n
an
(b ⫽
/ 0)
bn
(2x)4 ⫽ 24 ⴢ x 4 ⫽ 16x 4
冉冊
x
2
3
⫽
x3 x3
⫽
23 8
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9 䡲 PRECALCULUS REVIEW
The laws in Table 9.2 are valid for any real numbers a, b, m, and n whenever
the quantities are defined.
Remember, (x 2)3 ⬆ x 5. The correct equation is (x 2)3 ⫽ x 2⭈3 ⫽ x 6.
The next several examples illustrate the use of the laws of exponents.
EXAMPLE 1
Simplify the expressions:
a. (3x 2)(4x 3)
SOLUTION ✔
b.
165/4
161/2
d. (x 3y⫺2)⫺2
c. (62/3)3
a. (3x 2)(4x 3) ⫽ 12x 2⫹3 ⫽ 12x 5
b.
165/4
4
⫽ 165/4⫺1/2 ⫽ 163/4 ⫽ (兹
16)3 ⫽ 23 ⫽ 8
161/2
y 3/2
x 1/4
⫺2
⫽
y (3/2)(⫺2) y ⫺3 x 1/2
⫽
⫽ 3
x (1/4)(⫺2) x⫺1/2
y
y3/2
x 1/4
⫺2
(Law 2)
(Law 3)
d. (x 3y⫺2)⫺2 ⫽ (x 3)⫺2( y⫺2)⫺2 ⫽ x (3)(⫺2)y (⫺2)(⫺2) ⫽ x⫺6y 4 ⫽
冉 冊
冉 冊
(Law 1)
c. (62/3)3 ⫽ 62/3⭈3 ⫽ 66/3 ⫽ 62 ⫽ 36
e.
e.
y4
x6
(Law 4)
(Law 5)
䡲䡲䡲䡲
S IMPLIFYING R ADICALS
We can also use the laws of exponents to simplify expressions involving
radicals, as illustrated in the next example.
EXAMPLE 2
Simplify the expressions. (Assume x, y, and n are positive.)
b. 兹12m3n ⴢ 兹3m5n
4
16x 4y 8
a. 兹
SOLUTION ✔
c.
3
兹
⫺27x 6
3
兹
8y 3
4
a. 兹
16x 4y 8 ⫽ (16x 4y 8)1/4 ⫽ 161/4 ⴢ x 4/4y 8/4 ⫽ 2xy 2
b. 兹12m3n ⴢ 兹3m5n ⫽ 兹36m8n2 ⫽ (36m8n2)1/2 ⫽ 361/2 ⴢ m8/2n2/2 ⫽ 6m4n
c.
3
兹
⫺27x 6
兹8y 3
3
⫽
(⫺27x 6)1/3 ⫺271/3x 6/3
3x2
⫽ 1/3 3/3 ⫽ ⫺
3 1/3
(8y )
8 y
2y
䡲䡲䡲䡲
When a radical appears in the numerator or denominator of an algebraic
expression, we often try to simplify the expression by eliminating the radical
from the numerator or denominator. This process, called rationalization, is
illustrated in the next two examples.
9.1 䡲 E X P O N E N T S A N D R A D I C A L S
543
Rationalize the denominators:
EXAMPLE 3
3x
a.
SOLUTION ✔
b.
2 兹x
冪yx
3
a.
3x 兹x 3x 兹x 3x 兹x 3
3x
ⴢ
⫽
⫽
⫽ 兹x
⫽
2x
2
2 兹x 2 兹x 兹x 2 兹x 2
b.
冪 冪 冪
x
⫽
y
3
3
x
ⴢ
y
3
3
3
xy 2 兹
xy 2
y2 兹
⫽
⫽
3 3
y2
y
兹
y
䡲䡲䡲䡲
Rationalize the numerators:
EXAMPLE 4
SOLUTION ✔
冪yzx
a.
3 兹x
2x
a.
3 兹x 3 兹x 兹x 3 兹x 2
3x
3
ⴢ
⫽
⫽
⫽
⫽
2x
2x 兹x 2x 兹x 2x 兹x 2 兹x
b.
冪 冪
x2
yz 3
3
2
b.
3
3
3
3 3
兹
x
x
x
⫽ 3
⫽ 3
3
x 兹xyz
z 兹xy
䡲䡲䡲䡲
9.1 Exercises
In Exercises 1–16, evaluate the expression.
2. 8⫺4/3
1. 272/3
3.
5.
7.
冉 冊
冋冉 冊 册
冉 冊
1
0
兹3
1
8
1/3
7⫺5 ⴢ 72
7⫺2
⫺2
6.
⫺1
9. (1252/3)⫺1/2
11.
4. (71/2)4
兹32
兹8
165/8161/2
13.
167/8
15. 161/4 ⴢ (8)⫺1/3
8.
冋冉 冊 册
冉冊
1
3
⫺
9
16
18. 32 ⴢ 22 ⫽ 62
19. x 3 ⴢ 2x 2 ⫽ 2x 6
20. 33 ⫹ 3 ⫽ 34
21.
24x
⫽ 24x⫺3x
13x
22. (22 ⴢ 32)2 ⫽ 64
23.
1
1
⫽
4⫺3 64
24.
⫺1/2
25. (1.21/2)⫺1/2 ⫽ 1
12.
冪⫺8
27
14.
冉 冊
16.
17. x 4 ⫹ 2x 4 ⫽ 3x 4
⫺3
2
3 6
10. 兹
2
43/2 1
⫽
24 2
26. 52/3 ⴢ (25)2/3 ⫽ 25
In Exercises 27–32, rewrite the expression
using positive exponents only.
3
9⫺3 ⴢ 95
9⫺2
6 ⴢ6
6⫺1.4
2.5
In Exercises 17–26, determine whether the
statement is true or false. Give a reason for
your choice.
⫺1.9
⫺1/2
27. (xy)⫺2
29.
x⫺1/3
x1/2
31. 120(s ⫹ t)⫺3
28. 3s 1/3 ⴢ s ⫺7/3
30. 兹x⫺1 ⴢ 兹9x⫺3
32. (x ⫺ y)(x⫺1 ⫹ y⫺1)
9 䡲 PRECALCULUS REVIEW
544
In Exercises 33–48, simplify the expression.
33.
x 7/3
x ⫺2
34. (49x⫺2)⫺1/2
35. (x 2y⫺3)(x⫺5y 3)
37.
39.
x 3/4
x⫺1/4
38.
冉 冊
冉 冊冉 冊
x3
⫺27y⫺6
⫺3
41.
2
x
y⫺2
y
x
⫺2/3
40.
4
43. 兹x ⴢ 兹4x
3
5x 6y 3
36.
2x 2y 7
⫺2
冉 冊
冉 冊
5
⫺1/2
57.
59.
(r )
r 5⫺2n
61.
44. 兹81x y
6 ⫺4
3
48. 兹
27r 6 ⴢ 兹s 2t 4
9.2
50. 8
1/2
52. 6
1/2
3
58.
2 兹x
2y
60.
兹3y
1
62.
兹x
3
3
兹xy
5x 2
兹3x
冪2xy
In Exercises 63–68, rationalize the numerator
of the expression.
In Exercises 49–52, use the fact that 21/2 앒
1.414 and 31/2 앒 1.732 to evaluate the expression without using a calculator.
51. 9
56. (0.0001)⫺1/3
ex
e x⫺2
6
47. 兹64x 8y 3
3/4
55. 102.5
In Exercises 57–62, rationalize the denominator of the expression.
3 3a⫹b
46. 兹
x
49. 2
54. 10003/2
2
4
45. ⫺ 兹
16x 4y 8
3/2
53. 103/2
x 3y 2
z2
n 4
42.
In Exercises 53–56, use the fact that 101/2 앒
3.162 and 101/3 앒 2.154 to evaluate the expression without using a calculator.
63.
2 兹x
3
64.
兹x
24
65.
冪2yx
66.
2x
冪3y
67.
3 2
兹
xz
y
68.
兹x 2y
2x
3
3
3
Algebraic Expressions
O PERATIONS
WITH
A LGEBRAIC E XPRESSIONS
In calculus, we often work with algebraic expressions such as
2x 4/3 ⫺ x 1/3 ⫹ 1,
2x 2 ⫺ x ⫺
2
,
兹x
3xy ⫹ 2
,
x⫹1
2x 3 ⫹ 2x ⫹ 1
An algebraic expression of the form ax n, where the coefficient a is a real
number and n is a nonnegative integer, is called a monomial, meaning it
consists of one term. For example, 7x 2 is a monomial. A polynomial is a
monomial or the sum of two or more monomials. For example,
x 2 ⫹ 4x ⫹ 4,
x 3 ⫹ 5,
x 4 ⫹ 3x 2 ⫹ 3,
x 2y ⫹ xy ⫹ y
are all polynomials.
ADDING AND SUBTRACTING ALGEBRAIC EXPRESSIONS Constant terms
and terms containing the same variable factor are called like, or similar,
terms. Like terms may be combined by adding or subtracting their numerical
9.2 䡲 A L G E B R A I C E X P R E S S I O N S
545
coefficients. For example,
3x ⫹ 7x ⫽ 10x
and
7
1
xy ⫹ 3xy ⫽ xy
2
2
The distributive property of the real number system,
ab ⫹ ac ⫽ a ⴢ (b ⫹ c)
is used to justify this procedure.
To add or subtract two or more algebraic expressions, first remove the
parentheses and then combine like terms. The resulting expression is written
in order of decreasing degree from left to right.
嘷
R*
EXAMPLE 1
a. (3x 4 ⫹ 10x 3 ⫹ 6x 2 ⫹ 10x ⫹ 3) ⫹ (2x 4 ⫹ 10x 3 ⫹ 6x 2 ⫹ 4x)
⫽ 3x 4 ⫹ 2x 4 ⫹ 10x 3 ⫹ 10x 3 ⫹ 6x 2 ⫹ 6x 2 ⫹ 10x
⫹ 4x ⫹ 3
(Remove parentheses.)
4
3
⫽ 5x ⫹ 20x ⫹ 12x 2 ⫹ 14x ⫹ 3
(Combine like terms.)
b. (2x 4 ⫹ 3x 3 ⫹ 4x ⫹ 6) ⫺ (3x 4 ⫹ 9x 3 ⫹ 3x 2)
(Remove parentheses.)
⫽ 2x 4 ⫹ 3x 3 ⫹ 4x ⫹ 6 ⫺ 3x 4 ⫺ 9x 3 ⫺ 3x 2
4
4
3
3
2
⫽ 2x ⫺ 3x ⫹ 3x ⫺ 9x ⫺ 3x ⫹ 4x ⫹ 6
(Combine like terms.)
⫽ ⫺x 4 ⫺ 6x 3 ⫺ 3x 2 ⫹ 4x ⫹ 6
1
1
3
c. 2 兹x ⫹ 7y ⫺ 兹x ⫹ 3y ⫽ 2 兹x ⫺ 兹x ⫹ 7y ⫹ 3y ⫽ 兹x ⫹ 10y
2
2
2
d.
冉
冊 冉
冊
2
1
1
2x 3/2 ⫺ x 1/2 ⫹ x⫺1/3 ⫺ 3x 3/2 ⫹ x 1/2 ⫺ x ⫺1/3
3
2
3
2
1
1
⫽ 2x 3/2 ⫺ x 1/2 ⫹ x⫺1/3 ⫺ 3x 3/2 ⫺ x 1/2 ⫹ x⫺1/3
3
2
3
(Remove parentheses.)
1
2
1
⫽ 2x 3/2 ⫺ 3x 3/2 ⫺ x 1/2 ⫺ x 1/2 ⫹ x⫺1/3 ⫹ x⫺1/3
2
3
3
3
⫽ ⫺x 3/2 ⫺ x 1/2 ⫹ x⫺1/3
2
(Combine like terms.)
e. 2t 3 ⫺ 兵t 2 ⫺ [t ⫺ (2t ⫺ 1)] ⫹ 4其
⫽ 2t 3 ⫺ 兵t 2 ⫺ [t ⫺ 2t ⫹ 1] ⫹ 4其
⫽ 2t 3 ⫺ 兵t 2 ⫺ [⫺t ⫹ 1] ⫹ 4其
⫽ 2t 3 ⫺ 兵t 2 ⫹ t ⫺ 1 ⫹ 4其
⫽ 2t 3 ⫺ 兵t 2 ⫹ t ⫹ 3其
⫽ 2t 3 ⫺ t 2 ⫺ t ⫺ 3
(Remove parentheses and combine
like terms within brackets.)
(Remove brackets.)
(Combine like terms within braces.)
(Remove braces.)
䡲䡲䡲䡲
* The symbol 嘷
R indicates that these examples were selected from the calculus portion of the text
in order to help students review the algebraic computations they will actually be using in calculus.
546
9 䡲 PRECALCULUS REVIEW
An algebraic expression is said to be simplified if none of its terms are
similar. Observe that when the algebraic expression in (e) of Example 1 was
simplified, the innermost grouping symbols were removed first; that is, the
parentheses ( ) were removed first, the brackets [ ] second, and the braces
兵 其 third.
MULTIPLYING ALGEBRAIC EXPRESSIONS When we multiply algebraic expressions, each term of one algebraic expression is multiplied by each term
of the other. The resulting algebraic expression is then simplified.
EXAMPLE 2
Expand the expression (x ⫹ 3)(x ⫺ 2).
(x ⫹ 3)(x ⫺ 2) ⫽ x(x ⫺ 2) ⫹ 3(x ⫺ 2)
⫽ x 2 ⫺ 2x ⫹ 3x ⫺ 6
⫽ x2 ⫹ x ⫺ 6
SOLUTION ✔
嘷
R
EXAMPLE 3
SOLUTION ✔
䡲䡲䡲䡲
Perform the indicated operations:
a. (3x ⫺ 4)(3x 2 ⫺ 2x ⫹ 3)
b. (x 2 ⫹ 1)(3x 2 ⫹ 10x ⫹ 3)
c. (x 2 ⫹ x ⫹ 1)(x 2 ⫹ x ⫹ 1)
d. (e t ⫹ e⫺t )e t ⫺ e t(e t ⫺ e⫺t )
a. (3x ⫺ 4)(3x 2 ⫺ 2x ⫹ 3)
⫽ 3x(3x 2 ⫺ 2x ⫹ 3) ⫺ 4(3x 2 ⫺ 2x ⫹ 3)
⫽ 9x 3 ⫺ 6x 2 ⫹ 9x ⫺ 12x 2 ⫹ 8x ⫺ 12
⫽ 9x 3 ⫺ 18x 2 ⫹ 17x ⫺ 12
b. (x 2 ⫹ 1)(3x 2 ⫹ 10x ⫹ 3)
⫽ x 2(3x 2 ⫹ 10x ⫹ 3) ⫹ 1(3x 2 ⫹ 10x ⫹ 3)
⫽ 3x 4 ⫹ 10x 3 ⫹ 3x 2 ⫹ 3x 2 ⫹ 10x ⫹ 3
⫽ 3x 4 ⫹ 10x 3 ⫹ 6x 2 ⫹ 10x ⫹ 3
c. (x 2 ⫹ x ⫹ 1)(x 2 ⫹ x ⫹ 1)
⫽ x 2(x 2 ⫹ x ⫹ 1) ⫹ x(x 2 ⫹ x ⫹ 1) ⫹ 1(x 2 ⫹ x ⫹ 1)
⫽ x4 ⫹ x3 ⫹ x2 ⫹ x3 ⫹ x2 ⫹ x ⫹ x2 ⫹ x ⫹ 1
⫽ x 4 ⫹ 2x 3 ⫹ 3x 2 ⫹ 2x ⫹ 1
d. (e t ⫹ e⫺t )e t ⫺ e t (e t ⫺ e⫺t )
⫽ e 2t ⫹ e 0 ⫺ e 2t ⫹ e 0
⫽ e 2t ⫺ e 2t ⫹ e 0 ⫹ e 0
⫽1⫹1
⫽2
(Recall that e0 ⫽ 1.)
䡲䡲䡲䡲
Certain product formulas that are frequently used in algebraic computations are given in Table 9.3.
9.2 䡲 A L G E B R A I C E X P R E S S I O N S
547
Table 9.3 Product Formulas
Formula
Example
(a ⫹ b) ⫽ a ⫹ 2ab ⫹ b
2
2
(2x ⫹ 3y) ⫽ (2x)2 ⫹ 2(2x)(3y) ⫹ (3y)2
2
2
⫽ 4x 2 ⫹ 12xy ⫹ 9y 2
(a ⫺ b)2 ⫽ a 2 ⫺ 2ab ⫹ b2
(4x ⫺ 2y)2 ⫽ (4x)2 ⫺ 2(4x)(2y) ⫹ (2y)2
⫽ 16x 2 ⫺ 16xy ⫹ 4y 2
(a ⫹ b)(a ⫺ b) ⫽ a 2 ⫺ b 2
(2x ⫹ y)(2x ⫺ y) ⫽ (2x)2 ⫺ (y)2
⫽ 4x 2 ⫺ y 2
F ACTORING
Factoring is the process of expressing an algebraic expression as a product of
other algebraic expressions. For example, by applying the distributive property, we may write
3x 2 ⫺ x ⫽ x(3x ⫺ 1)
The first step in factoring an algebraic expression is to check whether it
contains any common terms. If it does, the greatest common term is then
factored out. For example, the common factor of the algebraic expression
2a2x ⫹ 4ax ⫹ 6a is 2a because
2a 2x ⫹ 4ax ⫹ 6a ⫽ 2a ⴢ ax ⫹ 2a ⴢ 2x ⫹ 2a ⴢ 3 ⫽ 2a(ax ⫹ 2x ⫹ 3)
嘷
R
EXAMPLE 4
Factor out the greatest common factor in each of the following expressions:
a. ⫺0.3t 2 ⫹ 3t
2
d. 2ye xy ⫹ 2xy 3e xy
SOLUTION ✔
b. 6a4b4c ⫺ 3a3b2c ⫺ 9a2b2
2
c. 2x 3/2 ⫺ 3x 1/2
e. 4x(x ⫹ 1)1/2 ⫺ 2x 2()(x ⫹ 1)⫺1/2
a. ⫺0.3t 2 ⫹ 3t ⫽ ⫺0.3t(t ⫺ 10)
b. 6a4b4c ⫺ 3a3b2c ⫺ 9a2b2 ⫽ 3a2b2(2a2b2c ⫺ ac ⫺ 3)
c. 2x 3/2 ⫺ 3x 1/2 ⫽ x 1/2(2x ⫺ 3)
2
2
2
d. 2ye xy ⫹ 2xy 3e xy ⫽ 2ye xy (1 ⫹ xy 2)
e. 4x(x ⫹ 1)1/2 ⫺ 2x 2()(x ⫹ 1)⫺1/2 ⫽ 4x(x ⫹ 1)1/2 ⫺ x 2(x ⫹ 1)⫺1/2
⫽ x(x ⫹ 1)⫺1/2[4(x ⫹ 1)1/2(x ⫹ 1)1/2 ⫺ x]
⫽ x(x ⫹ 1)⫺1/2[4(x ⫹ 1) ⫺ x]
⫽ x(x ⫹ 1)⫺1/2(4x ⫹ 4 ⫺ x) ⫽ x(x ⫹ 1)⫺1/2(3x ⫹ 4)
548
9 䡲 PRECALCULUS REVIEW
Here we select (x ⫹ 1)⫺1/2 as the common factor because it is ‘‘contained’’ in
each algebraic term. In particular, observe that
(x ⫹ 1)⫺1/2(x ⫹ 1)1/2(x ⫹ 1)1/2 ⫽ (x ⫹ 1)1/2
䡲䡲䡲䡲
Sometimes an algebraic expression may be factored by regrouping and
rearranging its terms and then factoring out a common term. This technique
is illustrated in Example 5.
EXAMPLE 5
Factor:
a. 2ax ⫹ 2ay ⫹ bx ⫹ by
SOLUTION ✔
b. 3x 兹y ⫺ 4 ⫺ 2 兹y ⫹ 6x
a. First, factor the common term 2a from the first two terms and the common
term b from the last two terms. Thus,
2ax ⫹ 2ay ⫹ bx ⫹ by ⫽ 2a(x ⫹ y) ⫹ b(x ⫹ y)
Since (x ⫹ y) is common to both terms of the polynomial, we may factor
it out. Hence,
2a(x ⫹ y) ⫹ b(x ⫹ y) ⫽ (x ⫹ y)(2a ⫹ b)
b. 3x 兹y ⫺ 4 ⫺ 2 兹y ⫹ 6x ⫽ 3x 兹y ⫺ 2 兹y ⫹ 6x ⫺ 4
⫽ 兹y(3x ⫺ 2) ⫹ 2(3x ⫺ 2)
⫽ (3x ⫺ 2)(兹y ⫹ 2)
䡲䡲䡲䡲
F ACTORING P OLYNOMIALS
A polynomial that is prime over a given set of numbers cannot be expressed
as a product of two polynomials of positive degree with coefficients belonging
to that set. For example, x 2 ⫹ 2x ⫹ 2 is prime over the set of polynomials
with integral coefficients because it cannot be expressed as the product of two
polynomials of positive degree that have integral coefficients.
The first step in factoring a polynomial is to find the common factors.
The next step is to express the polynomial as the product of a constant and/
or one or more prime polynomials.
Certain product formulas that are useful in factoring binomials and trinomials are listed in Table 9.4.
Table 9.4
Formula
Difference of two squares
x 2 ⫺ y 2 ⫽ (x ⫹ y)(x ⫺ y)
Example
x 2 ⫺ 36 ⫽ (x ⫹ 6)(x ⫺ 6)
8x 2 ⫺ 2y 2 ⫽ 2(4x 2 ⫺ y 2)
⫽ 2(2x ⫹ y)(2x ⫺ y)
9 ⫺ a 6 ⫽ (3 ⫹ a 3)(3 ⫺ a 3)
(continued)
9.2 䡲 A L G E B R A I C E X P R E S S I O N S
549
Table 9.4 (continued)
Formula
Example
Perfect-square trinomial
x 2 ⫹ 2xy ⫹ y 2 ⫽ (x ⫹ y)2
x 2 ⫹ 8x ⫹ 16 ⫽ (x ⫹ 4)2
x 2 ⫺ 2xy ⫹ y 2 ⫽ (x ⫺ y)2
4x 2 ⫺ 4xy ⫹ y 2 ⫽ (2x ⫺ y)2
Sum of two cubes
x 3 ⫹ y 3 ⫽ (x ⫹ y)(x 2 ⫺ xy ⫹ y 2)
z 3 ⫹ 27 ⫽ z 3 ⫹ (3)3
⫽ (z ⫹ 3)(z 2 ⫺ 3z ⫹ 9)
Difference of two cubes
x 3 ⫺ y 3 ⫽ (x ⫺ y)(x 2 ⫹ xy ⫹ y 2)
8x 3 ⫺ y 6 ⫽ (2x)3 ⫺ (y 2)3
⫽ (2x ⫺ y 2)(4x 2 ⫹ 2xy 2 ⫹ y 4)
The factors of the second-degree polynomial with integral coefficients
px 2 ⫹ qx ⫹ r
are (ax ⫹ b)(cx ⫹ d ), where ac ⫽ p, ad ⫹ bc ⫽ q, and bd ⫽ r. Since only a
limited number of choices are possible, we use a trial-and-error method to
factor polynomials having this form.
For example, to factor x 2 ⫺ 2x ⫺ 3, we first observe that the only possible
first-degree terms are
(x
)(x
)
(Since the coefficient of x 2 is 1)
Next, we observe that the product of the constant terms is (⫺3). This gives
us the following possible factors:
(x ⫺ 1)(x ⫹ 3)
(x ⫹ 1)(x ⫺ 3)
Looking once again at the polynomial x 2 ⫺ 2x ⫺ 3, we see that the coefficient
of x is ⫺2. Checking to see which set of factors yields ⫺2 for the coefficient
of x, we find that
Coefficients of inner terms
Factors
앗
Coefficients of outer terms
앗
(⫺1)(1) ⫹ (1)(3) ⫽ 2
Outer terms
앗
앗
(x ⫺ 1)(x ⫹ 3)
앖앖
Inner terms
Coefficients of inner terms
Coefficients of outer terms
앗
앗
(1)(1) ⫹ (1)(⫺3) ⫽ ⫺2
Outer terms
앗
앗
(x ⫹ 1)(x ⫺ 3)
앖앖
Inner terms
and we conclude that the correct factorization is
x 2 ⫺ 2x ⫺ 3 ⫽ (x ⫹ 1)(x ⫺ 3)
With practice, you will soon find that you can perform many of these steps
mentally and the need to write out each step will be eliminated.
550
9 䡲 PRECALCULUS REVIEW
嘷
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EXAMPLE 6
Factor:
a. 3x 2 ⫹ 4x ⫺ 4
SOLUTION ✔
b. 3x 2 ⫺ 6x ⫺ 24
a. Using trial and error, we find that the correct factorization is
3x 2 ⫹ 4x ⫺ 4 ⫽ (3x ⫺ 2)(x ⫹ 2)
b. Since each term has the common factor 3, we have
3x 2 ⫺ 6x ⫺ 24 ⫽ 3(x 2 ⫺ 2x ⫺ 8)
Using the trial-and-error method of factorization, we find that
x 2 ⫺ 2x ⫺ 8 ⫽ (x ⫺ 4)(x ⫹ 2)
Thus, we have
3x 2 ⫺ 6x ⫺ 24 ⫽ 3(x ⫺ 4)(x ⫹ 2)
R OOTS
OF
䡲䡲䡲䡲
P OLYNOMIAL E QUATIONS
A polynomial equation of degree n in the variable x is an equation of the
form
an x n ⫹ an⫺1 x n⫺1 ⫹ ⭈ ⭈ ⭈ ⫹ a0 ⫽ 0
where n is a nonnegative integer and a0 , a1 , . . . , an are real numbers with
an ⫽
/ 0. For example, the equation
⫺2x 5 ⫹ 8x 3 ⫺ 6x 2 ⫹ 3x ⫹ 1 ⫽ 0
is a polynomial equation of degree 5 in x.
The roots of a polynomial equation are precisely the values of x that
satisfy the given equation.* One way of finding the roots of a polynomial
equation is to first factor the polynomial and then solve the resulting equation.
For example, the polynomial equation
x 3 ⫺ 3x 2 ⫹ 2x ⫽ 0
may be rewritten in the form
x(x 2 ⫺ 3x ⫹ 2) ⫽ 0
x(x ⫺ 1)(x ⫺ 2) ⫽ 0
or
Since the product of two real numbers can be equal to zero if and only if one
(or both) of the factors is equal to zero, we have
x ⫽ 0,
x ⫺ 1 ⫽ 0,
or
x⫺2⫽0
from which we see that the desired roots are x ⫽ 0, 1, and 2.
* In this book, we are interested only in the real roots of an equation.
9.2 䡲 A L G E B R A I C E X P R E S S I O N S
551
T HE Q UADRATIC F ORMULA
In general, the problem of finding the roots of a polynomial equation is a
difficult one. But the roots of a quadratic equation (a polynomial equation
of degree 2) are easily found either by factoring or by using the following
quadratic formula.
Quadratic
Formula
The solutions of the equation ax 2 ⫹ bx ⫹ c ⫽ 0 (a ⫽
/ 0) are given by
x⫽
EXAMPLE 7
Solve each of the following quadratic equations:
a. 2x 2 ⫹ 5x ⫺ 12 ⫽ 0
SOLUTION ✔
⫺b ⫾ 兹b 2 ⫺ 4ac
2a
b. x 2 ⫽ ⫺3x ⫹ 8
a. The equation is in standard form, with a ⫽ 2, b ⫽ 5, and c ⫽ ⫺12. Using
the quadratic formula, we find
⫺b ⫾ 兹b2 ⫺ 4ac ⫺5 ⫾ 兹52 ⫺ 4(2)(⫺12)
⫽
2a
2(2)
⫺5 ⫾ 兹121 ⫺5 ⫾ 11
⫽
⫽
4
4
3
⫽ ⫺4 or
2
x⫽
This equation can also be solved by factoring. Thus,
2x 2 ⫹ 5x ⫺ 12 ⫽ (2x ⫺ 3)(x ⫹ 4) ⫽ 0
from which we see that the desired roots are x ⫽ 3/2 or x ⫽ ⫺4, as
obtained earlier.
b. We first rewrite the given equation in the standard form x 2 ⫹ 3x ⫺
8 ⫽ 0, from which we see that a ⫽ 1, b ⫽ 3, and c ⫽ ⫺8. Using the quadratic
formula, we find
x⫽
⫽
⫺b ⫾ 兹b2 ⫺ 4ac ⫺3 ⫾ 兹32 ⫺ 4(1)(⫺8)
⫽
2a
2(1)
⫺3 ⫾ 兹41
2
That is, the solutions are
⫺3 ⫹ 兹41
앒 1.7
2
and
⫺3 ⫺ 兹41
앒 ⫺4.7
2
In this case, the quadratic formula proves quite handy!
䡲䡲䡲䡲
9 䡲 PRECALCULUS REVIEW
552
9.2 Exercises
In Exercises 1–22, perform the indicated operations and simplify each expression.
1. (7x 2 ⫺ 2x ⫹ 5) ⫹ (2x 2 ⫹ 5x ⫺ 4)
3. (5y 2 ⫺ 2y ⫹ 1) ⫺ ( y 2 ⫺ 3y ⫺ 7)
4. 3(2a ⫺ b) ⫺ 4(b ⫺ 2a)
5. x ⫺ 兵2x ⫺ [⫺x ⫺ (1 ⫺ x)]其
6. 3x 2 ⫺ 兵x 2 ⫹ 1 ⫺ x[x ⫺ (2x ⫺ 1)]其 ⫹ 2
冉
冊 冉
冊
1
1
⫺ 1 ⫹ e ⫺ ⫺ ⫺ 1 ⫹ e⫺1
3
3
1
1
1
3
8. ⫺ y ⫺ x ⫹ 100 ⫹ x ⫹ y ⫺ 120
4
4
2
4
33. 4a 2 ⫺ b 2
34. 12x 2 ⫺ 3y 2
35. 10 ⫺ 14x ⫺ 12x 2
36. x 2 ⫺ 2x ⫺ 15
37. 3x 2 ⫺ 6x ⫺ 24
38. 3x 2 ⫺ 4x ⫺ 4
39. 12x 2 ⫺ 2x ⫺ 30
40. (x ⫹ y)2 ⫺ 1
41. 9x 2 ⫺ 16y 2
42. 8a 2 ⫺ 2ab ⫺ 6b2
43. x 6 ⫹ 125
44. x 3 ⫺ 27
In Exercises 45–52, perform the indicated operations and simplify the algebraic expression.
1
3
9. 3 兹8 ⫹ 8 ⫺ 2 兹y ⫹ 兹x ⫺ 兹y
2
4
10.
31. 6ac ⫹ 3bc ⫺ 4ad ⫺ 2bd
32. 3x 3 ⫺ x 2 ⫹ 3x ⫺ 1
2. (3x 2 ⫹ 5xy ⫹ 2y) ⫹ (4 ⫺ 3xy ⫺ 2x 2)
7.
In Exercises 31–44, factor each expression.
45. (x 2 ⫹ y 2)x ⫺ xy(2y)
16
16
8 2 2
x ⫹ x ⫹ x 2 ⫺ x ⫺ 2x ⫹ 2
9
3
3
3
46. 2kr(R ⫺ r) ⫺ kr 2
47. 2(x ⫺ 1)(2x ⫹ 2)3[4(x ⫺ 1) ⫹ (2x ⫹ 2)]
48. 5x 2(3x 2 ⫹ 1)4(6x) ⫹ (3x 2 ⫹ 1)5(2x)
11. (x ⫹ 8)(x ⫺ 2)
12. (5x ⫹ 2)(3x ⫺ 4)
13. (a ⫹ 5)2
14. (3a ⫺ 4b)2
49. 4(x ⫺ 1)2(2x ⫹ 2)3(2) ⫹ (2x ⫹ 2)4(2)(x ⫺ 1)
15. (x ⫹ 2y)2
16. (6 ⫺ 3x)2
50. (x 2 ⫹ 1)(4x 3 ⫺ 3x 2 ⫹ 2x) ⫺ (x 4 ⫺ x 3 ⫹ x 2)(2x)
17. (2x ⫹ y)(2x ⫺ y)
18. (3x ⫹ 2)(2 ⫺ 3x)
51. (x 2 ⫹ 2)2[5(x 2 ⫹ 2)2 ⫺ 3](2x)
19. (x 2 ⫺ 1)(2x) ⫺ x 2(2x)
20. (x1/2 ⫹ 1)
冉 冊
52. (x 2 ⫺ 4)(x 2 ⫹ 4)(2x ⫹ 8) ⫺ (x 2 ⫹ 8x ⫺ 4)(4x 3)
冉 冊
1 ⫺1/2
1
x
⫺ (x1/2 ⫺ 1) x⫺1/2
2
2
21. 2(t ⫹ 兹t)2 ⫺ 2t 2
In Exercises 53–58, find the real roots of each
equation by factoring.
22. 2x 2 ⫹ (⫺x ⫹ 1)2
In Exercises 23–30, factor out the greatest
common factor from each expression.
23. 4x 5 ⫺ 12x 4 ⫺ 6x 3
24. 4x 2y 2z ⫺ 2x 5y 2 ⫹ 6x 3y 2z 2
25. 7a ⫺ 42a b ⫹ 49a b
4
2
2
27. e⫺x ⫺ xe⫺x
⫺5/2
29. 2x
3
⫺ x⫺3/2
2
3
26. 3x
2/3
⫺ 2x
28. 2ye xy ⫹ 2xy 3 e xy
冉
1 2 3/2
u ⫺ 2u1/2
30.
2 3
54. 3x 2 ⫺ x ⫺ 4 ⫽ 0
55. 4t 2 ⫹ 2t ⫺ 2 ⫽ 0
56. ⫺6x 2 ⫹ x ⫹ 12 ⫽ 0
1
57. x 2 ⫺ x ⫹ 1 ⫽ 0
4
1
58. a 2 ⫹ a ⫺ 12 ⫽ 0
2
In Exercises 59–64, use the quadratic formula
to solve the quadratic equation.
1/3
2
53. x 2 ⫹ x ⫺ 12 ⫽ 0
2
冊
59. 4x 2 ⫹ 5x ⫺ 6 ⫽ 0
60. 3x 2 ⫺ 4x ⫹ 1 ⫽ 0
61. 8x 2 ⫺ 8x ⫺ 3 ⫽ 0
62. x 2 ⫺ 6x ⫹ 6 ⫽ 0
63. 2x 2 ⫹ 4x ⫺ 3 ⫽ 0
64. 2x 2 ⫹ 7x ⫺ 15 ⫽ 0
9.3 䡲 A L G E B R A I C F R A C T I O N S
9.3
553
Algebraic Fractions
R ATIONAL E XPRESSIONS
Quotients of polynomials are called rational expressions. Examples of rational
expressions are
6x ⫺ 1
,
2x ⫹ 3
3x 2y 3 ⫺ 2xy
,
4x
2
5ab
Since rational expressions are quotients in which the variables represent
real numbers, the properties of the real numbers apply to rational expressions
as well, and operations with rational fractions are performed in the same
manner as operations with arithmetic fractions. For example, using the properties of the real number system, we may write
a
ac a c a
⫽ ⴢ ⫽ ⴢ1⫽
bc b c b
b
where a, b, and c are any real numbers and b and c are not zero.
Similarly, using the same properties of real numbers, we may write
(x ⫹ 2)(x ⫺ 3) x ⫹ 2
⫽
(x ⫺ 2)(x ⫺ 3) x ⫺ 2
(x ⫽
/ 2, 3)
after ‘‘canceling’’ the common factors.
3兾 ⫹ 4x
/ 1 ⫹ 4x is an example of incorrect cancellation. Instead we need to
⫽
3兾
3 ⫹ 4x 3 4x
4x
write
⫽ ⫹ ⫽1⫹ .
3
3
3
3
A rational expression is simplified, or in lowest terms, when the numerator
and denominator have no common factors other than 1 and ⫺1 and the
expression contains no negative exponents.
嘷
R
EXAMPLE 1
SOLUTION ✔
Simplify the following expressions:
a.
x 2 ⫹ 2x ⫺ 3
x 2 ⫹ 4x ⫹ 3
c.
[(t 2 ⫹ 4)(2t ⫺ 4) ⫺ (t 2 ⫺ 4t ⫹ 4)(2t)]
(t 2 ⫹ 4)2
d.
(1 ⫹ x 2)2(⫺2x) ⫺ (1 ⫺ x 2)2(1 ⫹ x 2)(2x)
(1 ⫹ x 2)4
e.
(x ⫹ h)2 ⫺ x 2
h
a.
x 2 ⫹ 2x ⫺ 3 (x ⫹ 3)(x ⫺ 1) x ⫺ 1
⫽
⫽
x 2 ⫹ 4x ⫹ 3 (x ⫹ 3)(x ⫹ 1) x ⫹ 1
b.
3 ⫺ 4x ⫺ 4x 2
2x ⫺ 1
554
9 䡲 PRECALCULUS REVIEW
b.
c.
d.
e.
3 ⫺ 4x ⫺ 4x 2 (1 ⫺ 2x)(3 ⫹ 2x)
⫽
2x ⫺ 1
2x ⫺ 1
⫺(2x ⫺ 1)(2x ⫹ 3)
⫽
2x ⫺ 1
⫽ ⫺(2x ⫹ 3)
[1 ⫺ 2x ⫽ ⫺(2x ⫺ 1)]
[(t 2 ⫹ 4)(2t ⫺ 4) ⫺ (t 2 ⫺ 4t ⫹ 4)(2t)]
(t 2 ⫹ 4)2
3
2t ⫺ 4t 2 ⫹ 8t ⫺ 16 ⫺ 2t 3 ⫹ 8t 2 ⫺ 8t
⫽
(t 2 ⫹ 4)2
2
4t ⫺ 16
⫽ 2
(t ⫹ 4)2
4(t 2 ⫺ 4)
⫽ 2
(t ⫹ 4)2
(Carry out the indicated
multiplication.)
(Combine like terms.)
(Factor.)
(1 ⫹ x 2)2(⫺2x) ⫺ (1 ⫺ x 2)2(1 ⫹ x 2)(2x)
(1 ⫹ x 2)4
2
(1 ⫹ x )(2x)[(1 ⫹ x 2)(⫺1) ⫺ 2(1 ⫺ x 2)]
[Factor out
⫽
(1 ⫹ x 2)(2x).]
(1 ⫹ x 2)4
2x[⫺1 ⫺ x 2 ⫺ 2 ⫹ 2x 2 ]
[Divide numerator and
⫽
denominator by (1 ⫹ x 2).]
(1 ⫹ x 2)3
2x(x 2 ⫺ 3)
⫽
(Combine like terms.)
(1 ⫹ x 2)3
(x ⫹ h)2 ⫺ x 2 x 2 ⫹ xh ⫹ h 2 ⫺ x 2
⫽
h
h
h (x ⫹ h)
(Square the binomial and
⫽
combine like terms.)
h
1
1
(Divide numerator and
⫽ x⫹ h
denominator by h.)
2
4
䡲䡲䡲䡲
The operations of multiplication and division are performed with algebraic
fractions in the same manner as with arithmetic fractions (Table 9.5).
Table 9.5
Operation
Example
If P, Q, R, and S are polynomials, then
Multiplication
P R PR
ⴢ ⫽
Q S QS
(Q, S ⫽
/ 0)
2x (x ⫹ 1) 2x(x ⫹ 1) 2x 2 ⫹ 2x
ⴢ
⫽
⫽ 2
y (y ⫺ 1) y(y ⫺ 1)
y ⫺y
Division
PS
P R P S
⫼ ⫽ ⴢ ⫽
Q S Q R QR
(Q, R, S ⫽
/ 0)
x2 ⫹ 3 y2 ⫹ 1 x2 ⫹ 3
x
⫼
⫽
ⴢ 2
y
x
y
y ⫹1
⫽
x 3 ⫹ 3x
y3 ⫹ y
9.3 䡲 A L G E B R A I C F R A C T I O N S
555
When rational expressions are multiplied and divided, the resulting expressions should be simplified.
EXAMPLE 2
Perform the indicated operations and simplify:
a.
SOLUTION ✔
a.
b.
2x ⫺ 8 x 2 ⫹ 4x ⫹ 4
ⴢ
x⫹2
x 2 ⫺ 16
b.
x 2 ⫺ 6x ⫹ 9
x2 ⫺ 9
⫼ 2
3x ⫹ 12
6x ⫹ 18x
2x ⫺ 8 x 2 ⫹ 4x ⫹ 4
ⴢ
x⫹2
x 2 ⫺ 16
2(x ⫺ 4)
(x ⫹ 2)2
⫽
ⴢ
x ⫹ 2 (x ⫹ 4)(x ⫺ 4)
2(x ⫺ 4)(x ⫹ 2)(x ⫹ 2)
⫽
(x ⫹ 2)(x ⫹ 4)(x ⫺ 4)
2(x ⫹ 2)
⫽
x⫹4
x 2 ⫺ 6x ⫹ 9
x2 ⫺ 9
⫼ 2
3x ⫹ 12
6x ⫹ 18x
x 2 ⫺ 6x ⫹ 9 6x 2 ⫹ 18x
⫽
ⴢ
3x ⫹ 12
x2 ⫺ 9
2
(x ⫺ 3)
6x(x ⫹ 3)
⫽
ⴢ
3(x ⫹ 4) (x ⫹ 3)(x ⫺ 3)
(x ⫺ 3)(x ⫺ 3)(6x)(x ⫹ 3)
⫽
3(x ⫹ 4)(x ⫹ 3)(x ⫺ 3)
2x(x ⫺ 3)
⫽
x⫹4
[Cancel the common
factors (x ⫹ 2)(x ⫺ 4).]
[Cancel the common factors
3(x ⫹ 3)(x ⫺ 3).]
䡲䡲䡲䡲
For rational expressions, the operations of addition and subtraction are
performed by finding a common denominator of the fractions and then adding
or subtracting the fractions. Table 9.6 shows the rules for fractions with equal
denominators.
To add or subtract fractions that have different denominators, first find
a common denominator, preferably the least common denominator (LCD).
Then carry out the indicated operations following the procedure described in
Table 9.6.
Table 9.6
Operation
Example
If P, Q, and R are polynomials, then
Addition
P Q P⫹Q
⫹ ⫽
R R
R
(R ⫽
/ 0)
2x
6x
2x ⫹ 6x
8x
⫹
⫽
⫽
x⫹2 x⫹2
x⫹2
x⫹2
(R ⫽
/ 0)
3y
y
3y ⫺ y
2y
⫺
⫽
⫽
y⫺x y⫺x
y⫺x
y⫺x
Subtraction
P Q P⫺Q
⫺ ⫽
R R
R
556
9 䡲 PRECALCULUS REVIEW
To find the least common denominator (LCD) of two or more rational expressions,
1. Find the prime factors of each denominator.
2. Form the product of the different prime factors that occur in the denominators. Each prime factor in this product should be raised to the highest
power of that factor appearing in the denominators.
x
x x
⫽
/ ⫹ !
2⫹y 2 y
嘷
R
EXAMPLE 3
SOLUTION ✔
a.
a.
b.
3x ⫹ 4 4y ⫺ 2
⫹
4x
3y
b.
6(3x 2)
2x
⫹ 3
x ⫹1 x ⫹2
2
3x ⫹ 4 4y ⫺ 2
⫹
4x
3y
3x ⫹ 4 3y 4y ⫺ 2 4x
ⴢ ⫹
ⴢ
⫽
4x
3y
3y
4x
⫽
9xy ⫹ 12y 16xy ⫺ 8x
⫹
12xy
12xy
⫽
25xy ⫺ 8x ⫹ 12y
12xy
1
1
⫺
x⫹h x
[LCD ⫽ (4x)(3y) ⫽ 12xy]
2x
6(3x 2)
⫹ 3
x ⫹1 x ⫹2
2
2x(x 3 ⫹ 2) ⫹ 6(3x 2)(x 2 ⫹ 1)
(x 2 ⫹ 1)(x 3 ⫹ 2)
2x 4 ⫹ 4x ⫹ 18x 4 ⫹ 18x 2
⫽
(x 2 ⫹ 1)(x 3 ⫹ 2)
20x 4 ⫹ 18x 2 ⫹ 4x
⫽ 2
(x ⫹ 1)(x 3 ⫹ 2)
2x(10x 3 ⫹ 9x ⫹ 2)
⫽
(x 2 ⫹ 1)(x 3 ⫹ 2)
⫽
c.
c.
1
1
1
x 1 x⫹h
ⴢ ⫺ ⴢ
⫺ ⫽
x⫹h x x⫹h x x x⫹h
⫽
x⫹h
x
⫺
x(x ⫹ h) x(x ⫹ h)
⫽
x⫺x⫺h
x(x ⫹ h)
⫽
⫺h
x(x ⫹ h)
[LCD ⫽ (x 2 ⫹ 1)(x 3 ⫹ 2)]
[LCD ⫽ (x)(x ⫹ h)]
䡲䡲䡲䡲
9.3 䡲 A L G E B R A I C F R A C T I O N S
557
O THER A LGEBRAIC F RACTIONS
The techniques used to simplify rational expressions may also be used to
simplify algebraic fractions in which the numerator and denominator are not
polynomials, as illustrated in Example 4.
EXAMPLE 4
Simplify:
1
x⫹1
4
x⫺
x
1⫹
a.
SOLUTION ✔
b.
x⫺1 ⫹ y⫺1
x⫺2 ⫺ y⫺2
1
1
x⫹1
x⫹1⫹1
1ⴢ
⫹
x⫹1
x⫹1 x⫹1
x⫹1
⫽
⫽ 2
4
x 4
x ⫺4
x⫺
xⴢ ⫺
x
x x
x
1⫹
a.
⫽
x⫹2
x
x⫹2
x
ⴢ
ⴢ
⫽
x ⫹ 1 x 2 ⫺ 4 x ⫹ 1 (x ⫹ 2)(x ⫺ 2)
⫽
x
(x ⫹ 1)(x ⫺ 2)
1 1
y⫹x
⫹
x y
xy
x⫺1 ⫹ y⫺1
b. ⫺2
⫽
⫽
x ⫺ y⫺2 1
1 y2 ⫺ x2
⫺
x2 y2
x 2y 2
嘷
R
EXAMPLE 5
冉
x⫺n ⫽
冊
1
xn
⫽
y ⫹ x x 2y 2
y⫹x
(xy)2
ⴢ 2
⫽
ⴢ
2
xy y ⫺ x
xy (y ⫹ x)(y ⫺ x)
⫽
xy
y⫺x
䡲䡲䡲䡲
Show that the algebraic expression
8 4
3⫹ ⫺ 2
x x
4 5
2⫹ ⫺ 2
x x
is equal to
3x 2 ⫹ 8x ⫺ 4
2x 2 ⫹ 4x ⫺ 5
(We will work with the first form of this algebraic expression in Chapter 10
when we evaluate limits.)
558
9 䡲 PRECALCULUS REVIEW
SOLUTION ✔
The LCD of both the numerator and denominator is x 2. Therefore,
8 4 3(x 2) ⫹ 8(x) ⫺ 4
3⫹ ⫺ 2
x x
x2
⫽
2
4 5 2(x ) ⫹ 4(x) ⫺ 5
2⫹ ⫺ 2
x x
x2
3x 2 ⫹ 8x ⫺ 4
x2
⫽
ⴢ 2
2
x
2x ⫹ 4x ⫺ 5
2
3x ⫹ 8x ⫺ 4
⫽ 2
2x ⫹ 4x ⫺ 5
EXAMPLE 6
SOLUTION ✔
䡲䡲䡲䡲
Perform the given operations and simplify:
a.
x 2(2x 2 ⫹ 1)1/2 4x 3 ⫺ 6x 2 ⫹ x ⫺ 2
ⴢ
x⫺1
x(x ⫺ 1)(2x 2 ⫹ 1)
a.
x 2(2x 2 ⫹ 1)1/2 4x 3 ⫺ 6x 2 ⫹ x ⫺ 2
ⴢ
x⫺1
x(x ⫺ 1)(2x 2 ⫹ 1)
b.
12x 2
兹2x 2 ⫹ 3
⫹ 6 兹2x 2 ⫹ 3
x(4x 3 ⫺ 6x 2 ⫹ x ⫺ 2)
(x ⫺ 1)2(2x 2 ⫹ 1)1⫺1/2
x(4x 3 ⫺ 6x 2 ⫹ x ⫺ 2)
⫽
(x ⫺ 1)2(2x 2 ⫹ 1)1/2
⫽
b.
12x 2
兹2x 2 ⫹ 3
⫹ 6 兹2x 2 ⫹ 3 ⫽
12x 2
⫹ 6(2x 2 ⫹ 3)1/2
(2x 2 ⫹ 3)1/2
12x 2 ⫹ 6(2x 2 ⫹ 3)1/2(2x 2 ⫹ 3)1/2
(2x 2 ⫹ 3)1/2
2
12x ⫹ 6(2x 2 ⫹ 3)
⫽
(2x 2 ⫹ 3)1/2
24x 2 ⫹ 18
6(4x 2 ⫹ 3)
⫽
⫽
(2x 2 ⫹ 3)1/2 兹2x 2 ⫹ 3
⫽
䡲䡲䡲䡲
R ATIONALIZING A LGEBRAIC F RACTIONS
When the denominator of an algebraic fraction contains sums or differences
involving radicals, we may rationalize the denominator—that is, transform
the fraction into an equivalent one with a denominator that does not contain
radicals. In doing so, we make use of the fact that
(兹a ⫹ 兹b)(兹a ⫺ 兹b) ⫽ (兹a)2 ⫺ (兹b)2
⫽a⫺b
This procedure is illustrated in Example 7.
9.3 䡲 A L G E B R A I C F R A C T I O N S
Rationalize the denominator of
EXAMPLE 7
Upon multiplying
(1 ⫺ 兹x), we obtain
SOLUTION ✔
the
559
1
.
1 ⫹ 兹x
numerator
and
the
denominator
1 ⫺ 兹x
1
1
ⴢ
⫽
1 ⫹ 兹x 1 ⫹ 兹x 1 ⫺ 兹x
1 ⫺ 兹x
⫽
1 ⫺ (兹x)2
1 ⫺ 兹x
⫽
1⫺x
by
䡲䡲䡲䡲
In other situations, it may be necessary to rationalize the numerator of
an algebraic expression. In calculus, for example, one encounters the following problem.
Rationalize the numerator of
EXAMPLE 8
兹1 ⫹ h ⫺ 1
SOLUTION ✔
h
⫽
兹1 ⫹ h ⫺ 1
h
.
兹1 ⫹ h ⫺ 1 兹1 ⫹ h ⫹ 1
ⴢ
h
兹1 ⫹ h ⫹ 1
(兹1 ⫹ h) ⫺ (1)2
h(兹1 ⫹ h ⫹ 1)
1⫹h⫺1
⫽
h(兹1 ⫹ h ⫹ 1)
h
⫽
h(兹1 ⫹ h ⫹ 1)
1
⫽
兹1 ⫹ h ⫹ 1
2
⫽
[(兹1 ⫹ h)2 ⫽ 兹1 ⫹ h ⴢ 兹1 ⫹ h
⫽ 1 ⫹ h]
䡲䡲䡲䡲
9.3 EXERCISES
In Exercises 1–10, simplify the expression.
x ⫹x⫺2
x2 ⫺ 4
2
1.
12t 2 ⫹ 12t ⫹ 3
3.
4t 2 ⫺ 1
2a ⫺ 3ab ⫺ 9b
2ab 2 ⫹ 3b 3
2
2.
x 3 ⫹ 2x 2 ⫺ 3x
4.
⫺2x 2 ⫺ x ⫹ 3
5.
(4x ⫺ 1)(3) ⫺ (3x ⫹ 1)(4)
(4x ⫺ 1)2
6.
(1 ⫹ x 2)2(2) ⫺ 2x(2)(1 ⫹ x 2)(2x)
(1 ⫹ x 2)4
2
7.
(2x ⫹ 3)(1) ⫺ (x ⫹ 1)(2)
(2x ⫹ 3)2
8.
(x 2 ⫺ 1)4(2x) ⫺ x 2(4)(x 2 ⫺ 1)3(2x)
(x 2 ⫺ 1)8
9.
(e x ⫹ 1)e x ⫺ e x (2)(e x ⫹ 1)(e x )
(e x ⫹ 1)4
10.
(u 3 ⫺ 1)(3)(u 2 ⫹ u ⫺ 1)2(2u ⫹ 1) ⫺ (u 2 ⫹ u ⫺ 1)3(3u 2)
(u 3 ⫺ 1)2
9 䡲 PRECALCULUS REVIEW
560
In Exercises 11–34, perform the indicated operations and simplify each expression.
11.
2a 2 ⫺ 2b2
4a ⫹ 4b
ⴢ 2
b⫺a
a ⫹ 2ab ⫹ b 2
12.
x 2 ⫺ 6x ⫹ 9
3x ⫹ 6
ⴢ
x 2 ⫺ x ⫺ 6 2x 2 ⫺ 7x ⫹ 3
35. a.
3x 2 ⫹ 2x ⫺ 1
x2 ⫺ 1
⫼ 2
2x ⫹ 6
x ⫹ 2x ⫺ 3
14.
3x 2 ⫺ 4xy ⫺ 4y 2 (2y ⫺ x)2
⫼
x 2y
x 3y
15.
58
1
⫹
3(3t ⫹ 2) 3
19.
16.
21.
x⫹
1
(x 2 ⫺ 1)
x
20.
5
4
⫺
x 2 ⫺ 9 x 2 ⫺ 6x ⫹ 9
23. 2 ⫹
1
2a
⫺
a⫹2 a⫺2
22.
冉
2x ⫺
冊
x
2x ⫹ 3
⫹
1 ⫺ x x2 ⫺ 1
24. x ⫺
x
2
⫹
x⫹2 x⫺2
x⫺2 ⫺ y⫺2
27.
x⫹y
x⫺3 ⫺ y⫺3
28. ⫺1
x ⫺ y⫺1
29.
30.
2 兹2x 2 ⫹ 7
x3
兹8 ⫺ 3x 2
⫹ 兹2x ⫹ 7
(⫺6x) ⫹ 兹8 ⫺ 3x 2(6x 2)
a
1⫹
b
x
1 3
⫹
x x2
b.
1
1⫺
x
x4 ⫹ 1
37. a. 3
x ⫺1
1
x3
b.
1
1⫺ 3
x
38. a.
冪
4x 2 ⫹ 2x ⫺ 1
x2 ⫹ 2
x⫹
b.
2
1 1
⫹
x y
26.
1
1⫺
xy
2
b.
x⫹3
36. a. 2
x ⫺x
3
(x 3 ⫹ 1)
x
1
1⫹
x
25.
1
1⫺
x
4x 2
ax
x⫹b
a⫹1 b⫺2
⫹
3a
5b
⫺xe x
18.
⫹ ex
x⫹1
冉 冊
2(2x ⫺ 3)1/3 ⫺ (x ⫺ 1)(2x ⫺ 3)⫺2/3
(2x ⫺ 3)2/3
In Exercises 35–38, show that expressions (a)
and (b) are equivalent.
13.
2x
3x
17.
⫺
2x ⫺ 1 2x ⫹ 5
34.
冪
2 1
4⫹ ⫺ 2
x x
2
1⫹ 2
x
In Exercises 39–44, rationalize the denominator of each expression.
39.
41.
43.
1
兹3 ⫺ 1
1
兹x ⫺ 兹y
兹a ⫹ 兹b
兹a ⫺ 兹b
40.
42.
44.
1
兹x ⫹ 5
a
1 ⫺ 兹a
2 兹a ⫹ 兹b
2 兹a ⫺ 兹b
In Exercises 45–50, rationalize the numerator
of each expression.
31.
2x(x ⫹ 1)⫺1/2 ⫺ (x ⫹ 1)1/2
x2
45.
兹x
3
46.
3
兹
y
x
32.
(x 2 ⫹ 1)1/2 ⫺ 2x 2(x 2 ⫹ 1)⫺1/2
1 ⫺ x2
47.
1 ⫺ 兹3
3
48.
兹x ⫺ 1
x
33.
(2x ⫹ 1)1/2 ⫺ (x ⫹ 2)(2x ⫹ 1)⫺1/2
2x ⫹ 1
49.
50.
兹x ⫹ 3 ⫺ 兹x
3
1 ⫹ 兹x ⫹ 2
兹x ⫹ 2
9.4 䡲 I N E Q U A L I T I E S A N D A B S O L U T E V A L U E
9.4
561
Inequalities and Absolute Value
I NTERVALS
The system of real numbers and some of its properties were mentioned briefly
in Section 1.1. In much of our later work, we restrict our attention to certain
subsets of the set of real numbers. For example, if x denotes the number of
cars rolling off an assembly line each day in an automobile assembly plant,
then x must be nonnegative—that is, x ⱖ 0. Taking this example one step
further, suppose management decides that the daily production must not
exceed 200 cars. Then x must satisfy the inequality 0 ⱕ x ⱕ 200.
More generally, we will be interested in the following subsets of real
numbers: open intervals, closed intervals, and half-open intervals. The set of
all real numbers that lie strictly between two fixed numbers a and b is called
an open interval (a, b). It consists of all real numbers x that satisfy the
inequalities a ⬍ x ⬍ b, and it is called ‘‘open’’ because neither of its end
points is included in the interval. A closed interval contains both of its end
points. Thus, the set of all real numbers x that satisfy the inequalities a ⱕ x
ⱕ b is the closed interval [a, b]. Notice that square brackets are used to
indicate that the end points are included in this interval. Half-open intervals
contain only one of their end points. Thus, the interval [a, b) is the set of all
real numbers x that satisfy a ⱕ x ⬍ b, whereas the interval (a, b] is described
by the inequalities a ⬍ x ⱕ b. Examples of these finite intervals are illustrated
in Table 9.7.
Table 9.7 Finite Intervals
Interval
Open
Graph
x
(a, b)
a
b
a
b
Half-open (a, b]
a
b
Half-open [a, b)
a
b
Closed
[a, b]
Example
x
x
x
x
(⫺2, 1)
[⫺1, 2]
( , 3]
[⫺ , 3)
3
–3 –2 –1
0
1
2
–1
0
1
2
1
2
3
1
2
3
x
x
0
1
2
x
– 12 0
In addition to finite intervals, we encounter infinite intervals. Examples of infinite intervals are the half-lines (a, 앝), [a, 앝), (⫺앝, a), and (⫺앝, a]
defined by the set of all real numbers that satisfy x ⬎ a, x ⱖ a, x ⬍ a, and
x ⱕ a, respectively. The symbol 앝, called infinity, is not a real number. It is
used here only for notational purposes in conjunction with the definition of
infinite intervals. The notation (⫺앝, 앝) is used for the set of all real numbers
562
9 䡲 PRECALCULUS REVIEW
x since, by definition, the inequalities ⫺앝 ⬍ x ⬍ 앝 hold for any real number
x. Infinite intervals are illustrated in Table 9.8.
Table 9.8 Infinite Intervals
Interval
(a, 앝)
[a, 앝)
Graph
Example
x
a
x
a
x
(⫺앝, a)
a
(⫺앝, a]
a
x
P ROPERTIES
x
(2, 앝)
[⫺1, 앝)
0
2
x
–1
0
x
(⫺앝, 1)
(⫺앝, ⫺]
1
0
1
– 12 0
1
x
2
OF I NEQUALITIES
In practical applications, intervals are often found by solving one or more
inequalities involving a variable. In such situations, the following properties
may be used to advantage.
Properties of
Inequalities
If a, b, and c are any real numbers, then
Example
Property 1
If a ⬍ b and b ⬍ c,
then a ⬍ c.
2 ⬍ 3 and 3 ⬍ 8, so 2 ⬍ 8
Property 2
If a ⬍ b, then
a ⫹ c ⬍ b ⫹ c.
⫺5 ⬍ ⫺3, so ⫺5 ⫹ 2 ⬍ ⫺3 ⫹ 2;
that is, ⫺3 ⬍ ⫺1
Property 3
If a ⬍ b and c ⬎ 0,
then ac ⬍ bc.
⫺5 ⬍ ⫺3, and since 2 ⬎ 0, we
have (⫺5)(2) ⬍ (⫺3)(2); that is,
⫺10 ⬍ ⫺6
Property 4
If a ⬍ b and c ⬍ 0,
then ac ⬎ bc.
⫺2 ⬍ 4, and since ⫺3 ⬍ 0, we
have (⫺2)(⫺3) ⬎ (4)(⫺3); that
is, 6 ⬎ ⫺12
Similar properties hold if each inequality sign, ⬍, between a and b is replaced
by ⱖ, ⬎, or ⱕ.
A real number is a solution of an inequality involving a variable if a true
statement is obtained when the variable is replaced by that number. The set
of all real numbers satisfying the inequality is called the solution set.
9.4 䡲 I N E Q U A L I T I E S A N D A B S O L U T E V A L U E
EXAMPLE 1
SOLUTION ✔
563
Find the set of real numbers that satisfy ⫺1 ⱕ 2x ⫺ 5 ⬍ 7.
Add 5 to each member of the given double inequality, obtaining
4 ⱕ 2x ⬍ 12
Next, multiply each member of the resulting double inequality by 1/2, yielding
2ⱕx⬍6
Thus, the solution is the set of all values of x lying in the interval [2, 6).
䡲䡲䡲䡲
EXAMPLE 2
SOLUTION ✔
9.1
Sign diagram for (x ⫹ 4)(x ⫺ 2)
FIGURE
Solve the inequality x 2 ⫹ 2x ⫺ 8 ⬍ 0.
Observe that x 2 ⫹ 2x ⫺ 8 ⫽ (x ⫹ 4)(x ⫺ 2), so the given inequality is equivalent
to the inequality (x ⫹ 4)(x ⫺ 2) ⬍ 0. Since the product of two real numbers
is negative if and only if the two numbers have opposite signs, we solve the
inequality (x ⫹ 4)(x ⫺ 2) ⬍ 0 by studying the signs of the two factors x ⫹ 4
and x ⫺ 2. Now, x ⫹ 4 ⬎ 0 when x ⬎ ⫺4, and x ⫹ 4 ⬍ 0 when x ⬍ ⫺4.
Similarly, x ⫺ 2 ⬎ 0 when x ⬎ 2, and x ⫺ 2 ⬍ 0 when x ⬍ 2. These results
are summarized graphically in Figure 9.1.
Sign of
(x + 4) – – 0 + + + + + + + + + + + + + + + + +
(x – 2) – – – – – – – – – – – – – – 0 + + + + + +
x
–5 –4 –3 –2 –1
0
1
2
3
4
5
From Figure 9.1 we see that the two factors x ⫹ 4 and x ⫺ 2 have opposite
signs when and only when x lies strictly between ⫺4 and 2. Therefore, the
required solution is the interval (⫺4, 2).
䡲䡲䡲䡲
EXAMPLE 3
SOLUTION ✔
FIGURE
9.2
x⫹1
Sign diagram for
x⫺1
Solve the inequality
x⫹1
ⱖ 0.
x⫺1
The quotient (x ⫹ 1)/(x ⫺ 1) is strictly positive if and only if both the numerator
and the denominator have the same sign. The signs of x ⫹ 1 and x ⫺ 1 are
shown in Figure 9.2.
Sign of
(x + 1) – – – – – – 0 + + + + + + + + + +
(x – 1) – – – – – – – – – – 0 + + + + + +
x
–4 –3 –2 –1
0
1
2
3
4
564
9 䡲 PRECALCULUS REVIEW
From Figure 9.2 we see that x ⫹ 1 and x ⫺ 1 have the same sign when
and only when x ⬍ ⫺1 or x ⬎ 1. The quotient (x ⫹ 1)/(x ⫺ 1) is equal to
zero when and only when x ⫽ ⫺1. Therefore, the required solution is the set
of all x in the intervals (⫺앝, ⫺1] and (1, 앝).
䡲䡲䡲䡲
EXAMPLE 4
The management of Corbyco, a giant conglomerate, has estimated that x
thousand dollars is needed to purchase
100,000(⫺1 ⫹ 兹1 ⫹ 0.001x)
shares of common stock of the Starr Communications Company. Determine
how much money Corbyco needs in order to purchase at least 100,000 shares
of Starr’s stock.
SOLUTION ✔
The amount of cash Corbyco needs to purchase at least 100,000 shares is
found by solving the inequality
100,000(⫺1 ⫹ 兹1 ⫹ 0.001x) ⱖ 100,000
Proceeding, we find
⫺1 ⫹ 兹1 ⫹ 0.001x ⱖ 1
兹1 ⫹ 0.001x ⱖ 2
1 ⫹ 0.001x ⱖ 4
(Square both sides.)
0.001x ⱖ 3
x ⱖ 3000
䡲䡲䡲䡲
so Corbyco needs at least $3,000,000.
A BSOLUTE V ALUE
Absolute Value
The absolute value of a number a is denoted by 兩a兩 and is defined by
兩a兩 ⫽
再
a
if a ⱖ 0
⫺a if a ⬍ 0
Since ⫺a is a positive number when a is negative, it follows that the absolute
value of a number is always nonnegative. For example, 兩5兩 ⫽ 5 and 兩⫺5兩 ⫽
⫺(⫺5) ⫽ 5. Geometrically, 兩a兩 is the distance between the origin and the point
on the number line that represents the number a (Figure 9.3).
9.3
The absolute value of a number
FIGURE
|5|
|5|
|a|
|a|
x
–5
(a)
0
5
x
–a
(b)
0
a
9.4 䡲 I N E Q U A L I T I E S A N D A B S O L U T E V A L U E
Absolute Value
Properties
565
If a and b are any real numbers, then
Property 5
兩⫺a兩 ⫽ 兩a兩
Example
兩⫺3兩 ⫽ ⫺(⫺3) ⫽ 3 ⫽ 兩3兩
Property 6
兩ab兩 ⫽ 兩a兩 兩b兩
兩(2)(⫺3)兩 ⫽ 兩⫺6兩 ⫽ 6
Property 7
冏冏
Property 8
兩a ⫹ b兩 ⱕ 兩a兩 ⫹ 兩b兩
兩a兩
a
⫽
b
兩b兩
(b ⬆ 0)
冏 冏 冏冏
⫽ 兩2兩 兩⫺3兩
3 兩⫺3兩
(⫺3)
3
⫽
⫽ ⫽
4 兩⫺4兩
(⫺4)
4
兩8 ⫹ (⫺5)兩 ⫽ 兩3兩 ⫽ 3
ⱕ 兩8兩 ⫹ 兩⫺5兩
⫽ 13
Property 8 is called the triangle inequality.
Evaluate each expression:
EXAMPLE 5
a. 兩앟 ⫺ 5兩 ⫹ 3
SOLUTION ✔
b. 兩兹3苵 ⫺ 2兩 ⫹ 兩2 ⫺ 兹3苵兩
a. Since 앟 ⫺ 5 ⬍ 0, we see that 兩앟 ⫺ 5兩 ⫽ ⫺(앟 ⫺ 5). Therefore,
兩앟 ⫺ 5兩 ⫹ 3 ⫽ ⫺(앟 ⫺ 5) ⫹ 3 ⫽ 8 ⫺ 앟
b. Since 兹3苵 ⫺ 2 ⬍ 0, we see that 兩兹3苵 ⫺ 2兩 ⫽ ⫺(兹3苵 ⫺ 2). Next, observe that
2 ⫺ 兹3苵 ⬎ 0, so 兩2 ⫺ 兹3苵兩 ⫽ 2 ⫺ 兹3苵. Therefore,
兩兹3 ⫺ 2兩 ⫹ 兩2 ⫺ 兹3兩 ⫽ ⫺(兹3 ⫺ 2) ⫹ (2 ⫺ 兹3)
⫽ 4 ⫺ 2兹3 ⫽ 2(2 ⫺ 兹3)
Solve the inequalities 兩x兩 ⱕ 5 and 兩x兩 ⱖ 5.
EXAMPLE 6
SOLUTION ✔
FIGURE
䡲䡲䡲䡲
9.4
First, we consider the inequality 兩x兩 ⱕ 5. If x ⱖ 0, then 兩x兩 ⫽ x, so 兩x兩 ⱕ 5
implies x ⱕ 5 in this case. On the other hand, if x ⬍ 0, then 兩x兩 ⫽ ⫺x, so
兩x兩 ⱕ 5 implies ⫺x ⱕ 5 or x ⱖ ⫺5. Thus, 兩x兩 ⱕ 5 means ⫺5 ⱕ x ⱕ 5
(Figure 9.4a). To obtain an alternative solution, observe that 兩x兩 is the distance from the point x to zero, so the inequality 兩x兩 ⱕ 5 implies immediately
that ⫺5 ⱕ x ⱕ 5.
[
–5
(a)
0
[
5
x
[
–5
0
[
5
x
(b)
Next, the inequality 兩x兩 ⱖ 5 states that the distance from x to zero is
greater than or equal to 5. This observation yields the result x ⱖ 5 or x ⱕ ⫺5
(Figure 9.4b).
䡲䡲䡲䡲
9 䡲 PRECALCULUS REVIEW
566
Solve the inequality 兩2x ⫺ 3兩 ⱕ 1.
EXAMPLE 7
SOLUTION ✔
9.5
兩2x ⫺ 3兩 ⱕ 1
The inequality 兩2x ⫺ 3兩 ⱕ 1 is equivalent to the inequalities ⫺1 ⱕ 2x ⫺ 3 ⱕ
1 (see Example 6). Thus, 2 ⱕ 2x ⱕ 4 and 1 ⱕ x ⱕ 2. The solution is therefore
given by the set of all x in the interval [1, 2] (Figure 9.5).
FIGURE
[
1
0
x
[
2
䡲䡲䡲䡲
9.4 Exercises
In Exercises 1–4, determine whether the
statement is true or false.
1. ⫺3 ⬍ ⫺20
2. ⫺5 ⱕ ⫺5
3.
2 5
⬎
3 6
5
11
4. ⫺ ⬍ ⫺
6
12
27.
x⫺2
ⱕ2
x⫺1
5. (3, 6)
6. (⫺2, 5]
7. [⫺1, 4)
6 1
8. [⫺ , ⫺ ]
5 2
9. (0, 앝)
10. (⫺앝, 5]
In Exercises 11–28, find the values of x that
satisfy the inequality (inequalities).
2x ⫺ 1
ⱕ4
x⫹2
In Exercises 29–38, evaluate the given expression.
29. 兩⫺6 ⫹ 2兩
In Exercises 5–10, show the given interval on
a number line.
28.
31.
兩⫺12 ⫹ 4兩
兩16 ⫺ 12兩
30. 4 ⫹ 兩⫺4兩
32.
冏
0.2 ⫺ 1.4
1.6 ⫺ 2.4
冏
33. 兹3兩⫺2兩 ⫹ 3兩⫺ 兹3兩
34. 兩⫺1兩 ⫹ 兹2兩⫺2兩
35. 兩앟 ⫺ 1兩 ⫹ 2
36. 兩앟 ⫺ 6兩 ⫺ 3
37. 兩 兹2 ⫺ 1兩 ⫹ 兩3 ⫺ 兹2兩
38. 兩2 兹3 ⫺ 3兩 ⫺兩 兹3 ⫺ 4兩
11. 2x ⫹ 4 ⬍ 8
12. ⫺6 ⬎ 4 ⫹ 5x
13. ⫺4x ⱖ 20
14. ⫺12 ⱕ ⫺3x
In Exercises 39–44, suppose that a and b are
real numbers other than zero and that a > b.
State whether the inequality is true or false.
15. ⫺6 ⬍ x ⫺ 2 ⬍ 4
16. 0 ⱕ x ⫹ 1 ⱕ 4
39. b ⫺ a ⬎ 0
40.
a
⬎1
b
41. a 2 ⬎ b 2
42.
1 1
⬎
a b
43. a 3 ⬎ b 3
44. ⫺a ⬍ ⫺b
17. x ⫹ 1 ⬎ 4 or x ⫹ 2 ⬍ ⫺1
18. x ⫹ 1 ⬎ 2 or x ⫺ 1 ⬍ ⫺2
19. x ⫹ 3 ⬎ 1 and x ⫺ 2 ⬍ 1
20. x ⫺ 4 ⱕ 1 and x ⫹ 3 ⬎ 2
21. (x ⫹ 3)(x ⫺ 5) ⱕ 0
22. (2x ⫺ 4)(x ⫹ 2) ⱖ 0
In Exercises 45–50, determine whether the
statement is true for all real numbers a and b.
23. (2x ⫺ 3)(x ⫺ 1) ⱖ 0
24. (3x ⫺ 4)(2x ⫹ 2) ⱕ 0
45. 兩⫺a兩 ⫽ a
46. 兩b 2 兩 ⫽ b 2
47. 兩a ⫺ 4兩 ⫽ 兩4 ⫺ a兩
48. 兩a ⫹ 1兩 ⫽ 兩a兩 ⫹ 1
49. 兩a ⫹ b兩 ⫽ 兩a兩 ⫹ 兩b兩
50. 兩a ⫺ b兩 ⫽ 兩a兩 ⫺ 兩b兩
25.
x⫹3
ⱖ0
x⫺2
26.
2x ⫺ 3
ⱖ4
x⫹1
CHAPTER 9 䡲 S U M M A R Y A N D R E V I E W
51. D RIVING R ANGE OF A C AR An advertisement for a certain
car states that the EPA fuel economy is 20 mpg city and
27 mpg highway and that the car’s fuel-tank capacity is
18.1 gal. Assuming ideal driving conditions, determine
the driving range for the car from these data.
52. Find the minimum cost C (in dollars), given that
5(C ⫺ 25) ⱖ 1.75 ⫹ 2.5C
53. Find the maximum profit P (in dollars) given that
6(P ⫺ 2500) ⱕ 4(P ⫹ 2400)
54. C ELSIUS AND F AHRENHEIT T EMPERATURES The relationship
between Celsius (⬚C) and Fahrenheit (⬚F) temperatures
is given by the formula
57. Q UALITY C ONTROL The PAR Manufacturing Company
manufactures steel rods. Suppose the rods ordered by a
customer are manufactured to a specification of 0.5 in.
and are acceptable only if they are within the tolerance
limits of 0.49 in. and 0.51 in. Letting x denote the diameter of a rod, write an inequality using absolute value to
express a criterion involving x that must be satisfied in
order for a rod to be acceptable.
58. Q UALITY C ONTROL The diameter x (in inches) of a batch of
ball bearings manufactured by the PAR Manufacturing
Company satisfies the inequality
兩x ⫺ 0.1兩 ⱕ 0.01
What is the smallest diameter a ball bearing in the batch
can have? The largest diameter?
59. M EETING P ROFIT G OALS A manufacturer of a certain commodity has estimated that her profit in thousands of
dollars is given by the expression
5
C ⫽ (F ⫺ 32)
9
a. If the temperature range for Montreal during
the month of January is ⫺15⬚ ⬍ ⬚C ⬍ ⫺5⬚, find the
range in degrees Fahrenheit in Montreal for the same
period.
b. If the temperature range for New York City during
the month of June is 63⬚ ⬍ ⬚F ⬍ 80⬚, find the range in
degrees Celsius in New York City for the same period.
55. M EETING S ALES T ARGETS A salesman’s monthly commission is 15% on all sales over $12,000. If his goal is to
make a commission of at least $3000 per month, what
minimum monthly sales figures must he attain?
56. M ARKUP ON A C AR The markup on a used car was at least
30% of its current wholesale price. If the car was sold
for $2800, what was the maximum wholesale price?
C HAPTER
567
⫺6x 2 ⫹ 30x ⫺ 10
where x (in thousands) is the number of units produced.
What production range will enable the manufacturer to
realize a profit of at least $14,000 on the commodity?
60. D ISTRIBUTION OF I NCOMES The distribution of income in a
certain city can be described by the exponential model
y ⫽ (2.8 ⴢ 1011)(x)⫺1.5, where y is the number of families
with an income of x or more dollars.
a. How many families in this city have an income of
$20,000 or more?
b. How many families have an income of $40,000 or
more?
c. How many families have an income of $100,000 or
more?
9 Summary of Principal Formulas and Terms
Formulas
1. Product formulas
(a ⫹ b)2 ⫽ a 2 ⫹ 2ab ⫹ b 2
(a ⫺ b)2 ⫽ a 2 ⫺ 2ab ⫹ b 2
(a ⫹ b)(a ⫺ b) ⫽ a 2 ⫺ b 2
2. Quadratic formula
x⫽
⫺b ⫾ 兹b 2 ⫺ 4ac
2a
9 䡲 PRECALCULUS REVIEW
568
Terms
C HAPTER
roots of a polynomial equation
finite interval
open interval
infinite interval
closed interval
absolute value
half-open interval
triangle inequality
9 R EVIEW E XERCISES
In Exercises 1–6, evaluate the expression.
1.
冉冊
9
4
3/2
2.
3. (3 ⴢ 4)
⫺2
4. (⫺8)
(3 ⴢ 2 )(4 ⴢ 3 )
2 ⴢ 93
⫺3
5.
56
54
5
6.
3
3 兹54
11.
兹81xyz
4
10. (2x 3)(⫺3x⫺2)
5
冉 冊冉 冊
3xy 2
4x 3y
⫺2
3xy 3
2x 2
冉 冊
1 ⫺1/2
x
6
24.
⫺2x
⫹ 4 兹x ⫹ 1
兹x ⫹ 1
冉
冊 冉
冊
In Exercises 25–28, solve the equation by factoring.
25. 8x 2 ⫹ 2x ⫺ 3 ⫽ 0
26. ⫺6x 2 ⫺ 10x ⫹ 4 ⫽ 0
27. ⫺x 3 ⫺ 2x 2 ⫹ 3x ⫽ 0
28. 2x 4 ⫹ x 2 ⫽ 1
In Exercises 29–32, find the values of x that
satisfy the given inequalities.
29. ⫺x ⫹ 3 ⱕ 2x ⫹ 9
3
3
3
13. 兹81x 5y 10 兹
9xy 2
2
4x
3
⫹3
3 2x 2 ⫺ 1
3x ⫺ 1)
3
a 6 b⫺5
8. 3 ⫺2 ⫺3
(a b )
兹16x 5yz
23.
兹18
4
9.
6x
1
⫹
2(3x 2 ⫹ 2) 4(x ⫹ 2)
5/3
In Exercises 7–14, simplify the expression.
4(x 2 ⫹ y)3
7.
x2 ⫹ y
22.
2
⫺1 ⫺2 ⫺1
3 2
12. (⫺3a b ) (2a b )
14.
冉
冊
⫺x 1/2y 2/3
x 1/3y 3/4
6
30. ⫺2 ⱕ 3x ⫹ 1 ⱕ 7
31. x ⫺ 3 ⬎ 2 or x ⫹ 3 ⬍ ⫺1
32. 2x 2 ⬎ 50
In Exercises 33–36, evaluate the expression.
In Exercises 15–20, factor the expression.
15. ⫺2앟 2r 3 ⫹ 100앟r 2
16. 2v 3w ⫹ 2vw 3 ⫹ 2u 2vw
17. 16 ⫺ x 2
18. 12t 3 ⫺ 6t 2 ⫺ 18t
19. ⫺2x 2 ⫺ 4x ⫹ 6
20. 12x 2 ⫺ 92x ⫹ 120
In Exercises 21–24, perform the indicated operations and simplify the expression.
21.
(t ⫹ 6)(60) ⫺ (60t ⫹ 180)
(t ⫹ 6)2
33. 兩⫺5 ⫹ 7兩 ⫹ 兩⫺2兩
34.
冏
5 ⫺ 12
⫺4 ⫺ 3
冏
35. 兩2앟 ⫺ 6兩 ⫺ 앟
36. 兩 兹3 ⫺ 4兩 ⫹ 兩4 ⫺ 2 兹3兩
In Exercises 37–40, find the value(s) of x that
satisfy the expression.
37. 2x 2 ⫹ 3x ⫺ 2 ⱕ 0
38.
1
⬎2
x⫹2
CHAPTER 9 䡲 S U M M A R Y A N D R E V I E W
39. 兩2x ⫺ 3兩 ⬍ 5
40.
冏 冏
41. Rationalize the numerator:
兹x ⫺ 1
x⫺1
42. Rationalize the denominator:
兹x ⫺ 1
2 兹x
x⫹1
⫽5
x⫺1
569
In Exercises 43 and 44, use the quadratic formula to solve the quadratic equation.
43. x 2 ⫺ 2x ⫺ 5 ⫽ 0
44. 2x 2 ⫹ 8x ⫹ 7 ⫽ 0
45. Find the minimum cost C (in dollars), given that
2(1.5C ⫹ 80) ⱕ 2(2.5C ⫺ 20)
46. Find the maximum revenue R (in dollars), given that
12(2R ⫺ 320) ⱕ 4(3R ⫹ 240)