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CHAPTER 0
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A Precalculus Review
ABSOLUTE VALUE AND DISTANCE ON THE REAL NUMBER LINE
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■
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Find the absolute values of real numbers and understand the properties of absolute value.
Find the distance between two numbers on the real number line.
Define intervals on the real number line.
Find the midpoint of an interval and use intervals to model and solve real-life problems.
Absolute Value of a Real Number
T E C H N O L O G Y
Absolute value expressions
can be evaluated on a
graphing utility. When an expression such as 3 8 is evaluated,
parentheses should surround the
expression, as in abs3 8.
Definition of Absolute Value
The absolute value of a real number a is
a a,
a,
if a ≥ 0
if a < 0.
At first glance, it may appear from this definition that the absolute value of a real
number can be negative, but this is not possible. For example, let a 3. Then,
because 3 < 0, you have
a 3
3
3.
The following properties are useful for working with absolute values.
Properties of Absolute Value
1. Multiplication:
2. Division:
3. Power:
4. Square root:
ab ab
a
a
, b 0
b
b
n
a an
a2 a
Be sure you understand the fourth property in this list. A common error in
algebra is to imagine that by squaring a number and then taking the square root,
you come back to the original number. But this is true only if the original number
is nonnegative. For instance, if a 2, then
22 4 2
but if a 2, then
22 4 2.
The reason for this is that (by definition) the square root symbol denotes only
the nonnegative root.
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SECTION 0.2
Absolute Value and Distance on the Real Number Line
0-9
Distance on the Real Number Line
Consider two distinct points on the real number line, as shown in Figure 0.9.
Directed distance
from a to b:
1. The directed distance from a to b is b a.
2. The directed distance from b to a is a b.
b
a
x
b−a
3. The distance between a and b is a b or b a .
In Figure 0.9, note that because b is to the right of a, the directed distance
from a to b (moving to the right) is positive. Moreover, because a is to the left of
b, the directed distance from b to a (moving to the left) is negative. The distance
between two points on the real number line can never be negative.
Directed distance
from b to a:
b
a
x
a−b
Distance between
a and b:
a
Distance Between Two Points on the Real Number Line
a − b or b − a
The distance d between points x1 and x2 on the real number line is
given by
FIGURE 0.9
d x2 x1 x2 x12 .
Note that the order of subtraction with x1 and x2 does not matter because
x2 x1 x1 x2
EXAMPLE 1
x2 x12 x1 x2 2.
and
Finding Distance on the Real Number Line
Determine the distance between 3 and 4 on the real number line. What is the
directed distance from 3 to 4? What is the directed distance from 4 to 3?
The distance between 3 and 4 is given by
SOLUTION
3 4 7 7
or
4 3 7 7
a b
or b a
as shown in Figure 0.10.
Distance = 7
x
−4 −3 −2 −1
0
1
2
3
4
5
FIGURE 0.10
The directed distance from 3 to 4 is
4 3 7.
ba
The directed distance from 4 to 3 is
3 4 7.
TRY
IT
x
b
ab
1
Determine the distance between 2 and 6 on the real number line. What is the
directed distance from 2 to 6? What is the directed distance from 6 to 2?
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A Precalculus Review
CHAPTER 0
Intervals Defined by Absolute Value
EXAMPLE 2
Defining an Interval on the Real Number Line
Find the interval on the real number line that contains all numbers that lie no more
than two units from 3.
SOLUTION Let x be any point in this interval. You need to find all x such that
the distance between x and 3 is less than or equal to 2. This implies that
x 3 ≤ 2.
Requiring the absolute value of x 3 to be less than or equal to 2 means that
x 3 must lie between 2 and 2. So, you can write
2 ≤ x 3 ≤ 2.
x − 3 ≤ 2
2 units
2 units
Solving this pair of inequalities, you have
x
0
1
2
3
4
5
2 3 ≤ x 3 3 ≤ 2 3
1 ≤
6
x
≤ 5.
Solution set
So, the interval is 1, 5 , as shown in Figure 0.11.
FIGURE 0.11
TRY
IT
2
Find the interval on the real number line that contains all numbers that lie no
more than four units from 6.
Two Basic Types of Inequalities Involving Absolute Value
Let a and d be real numbers, where d > 0.
x a ≤ d if and only if a d ≤ x ≤ a d.
x a ≥ d if and only if x ≤ a d or a d ≤ x.
Inequality
ALGEBRA
REVIEW
Be sure you see that inequalities
of the form x a ≥ d have solution sets consisting of two intervals. To describe the two intervals
without using absolute values, you
must use two separate inequalities,
connected by an “or” to indicate
union.
x a ≤ d
x a ≥ d
Interpretation
All numbers x
whose distance
from a is less than
or equal to d.
All numbers x
whose distance
from a is greater
than or equal to d.
Graph
d
d
x
a−d
a+d
a
d
d
x
a−d
a
a+d
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SECTION 0.2
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Absolute Value and Distance on the Real Number Line
Application
EXAMPLE 3
Quality Control
A large manufacturer hired a quality control firm to determine the reliability
of a product. Using statistical methods, the firm determined that the manufacturer
could expect 0.35% ± 0.17% of the units to be defective. If the manufacturer
offers a money-back guarantee on this product, how much should be budgeted to
cover the refunds on 100,000 units? (Assume that the retail price is $8.95.)
SOLUTION Let r represent the percent of defective units (written in decimal
form). You know that r will differ from 0.0035 by at most 0.0017.
0.0035 0.0017 ≤ r ≤ 0.0035 0.0017
0.0018 ≤ r ≤ 0.0052
Figure 0.12(a)
0.0018100,000 ≤ 100,000r ≤ 0.0052100,000
180 ≤
≤ 520.
Figure 0.12(b)
x
Finally, letting C be the cost of refunds, you have C 8.95x. So, the total cost of
refunds for 100,000 units should fall within the interval given by
TRY
IT
0.0052
r
0
Now, letting x be the number of defective units out of 100,000, it follows that
x 100,000r and you have
1808.95 ≤ 8.95x ≤ 5208.95
$1611 ≤ C ≤ $4654.
0.0018
0.002
0.004
0.006
(a) Percent of defective units
180
520
x
0
100 200 300 400 500 600
(b) Number of defective units
1611
4654
C
0
1000 2000 3000 4000 5000
(c) Cost of refunds
Figure 0.12(c)
FIGURE 0.12
3
Use the information in Example 3 to determine how much should be budgeted to cover refunds on 250,000 units.
In Example 3, the manufacturer should expect to spend between $1611 and
$4654 for refunds. Of course, the safer budget figure for refunds would be the
higher of these estimates. However, from a statistical point of view, the most
representative estimate would be the average of these two extremes. Graphically,
the average of two numbers is the midpoint of the interval with the two numbers
as endpoints, as shown in Figure 0.13.
Midpoint of an Interval
The midpoint of the interval with endpoints a and b is found by taking
the average of the endpoints.
Midpoint ab
2
Midpoint = 1611 +2 4654 = 3132.5
1611
4654
C
0
1000 2000 3000 4000 5000
FIGURE 0.13
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CHAPTER 0
E X E R C I S E S
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In Exercises 1–6, find (a) the directed distance from a to b, (b) the
directed distance from b to a, and (c) the distance between a
and b.
1. a 126, b 75
2. a 126, b 75
3. a 9.34, b 5.65
4. a 2.05, b 4.25
16
112
5. a 5 , b 75
18
61
6. a 5 , b 15
In Exercises 7–18, use absolute values to describe the given interval (or pair of intervals) on the real number line.
7. 2, 2
8. 3, 3
9. , 2 2, 10. , 3 3, 11. 2, 6
12. 7, 1
13. , 0 4, 14. , 20 24, 15. All numbers less than two units from 4
16. All numbers more than six units from 3
17. y is at most two units from a.
18. y is less than h units from c.
In Exercises 19–34, solve the inequality and sketch the graph of
the solution on the real number line.
5x > 10
3x 1 ≥ 4
2x 1 < 5
25 x ≥ 20
19. x < 5
20. 2x < 6
x
> 3
21.
2
22.
23. x 2 < 5
24.
x3
≥ 5
2
26.
25.
27. 10 x > 4
28.
29. 9 2x < 1
30. 1 31. x a ≤ b, b > 0
32.
33.
1 3
39. 2, 4 h 68.5
≤ 1
2.7
where h is measured in inches. Determine the interval on
the real number line in which these heights lie.
44. Biology The American Kennel Club has developed
guidelines for judging the features of various breeds of
dogs. For collies, the guidelines specify that the weights for
males satisfy the inequality
w 57.5
≤ 1
7.5
where w is measured in pounds. Determine the interval on
the real number line in which these weights lie.
45. Production The estimated daily production x at a refinery is given by
x 200,000 ≤ 25,000
where x is measured in barrels of oil. Determine the high
and low production levels.
46. Manufacturing The acceptable weights for a 20-ounce
cereal box are given by
x 20 ≤ 0.75
Item
Budget
Expense
47. Utilities
$4750.00
$5116.37
36. 8.6, 11.4
48. Insurance
$15,000.00
$14,695.00
38. 4.6, 1.3
49. Maintenance
$20,000.00
$22,718.35
50. Taxes
$7500.00
$8691.00
In Exercises 35–40, find the midpoint of the given interval.
37. 6.85, 9.35
notation to represent the two intervals in which expenses must
lie if they are to be within $500 and within 5% of the specified
budget amount and (b) using the more stringent constraint,
determine whether the given expense is at variance with the
budget restriction.
b > 0
5x
> b, b > 0
2
35. 7, 21
43. Statistics The heights h of two-thirds of the members of
a population satisfy the inequality
Budget Variance In Exercises 47–50, (a) use absolute value
3x a
< 2b, b > 0
4
34. a 42. Stock Price A stock market analyst predicts that over
the next year the price p of a stock will not change from its
current price of $3318 by more than $2. Use absolute values
to write this prediction as an inequality.
where x is measured in ounces. Determine the high and low
weights for the cereal box.
2x
< 1
3
2x a ≥ b,
41. Chemistry Copper has a melting point M within 0.2°C
of 1083.4°C. Use absolute values to write the range as an
inequality.
40.
56, 52