Download Inequalities and Absolute Value

1.3
Inequalities and Absolute Value
1.3
OBJECTIVES
1. Use the notation of inequalities
2. Graph inequalities
3. Use the absolute value notation
Let’s now consider two relations on the set of real numbers. These are the relations of order
or inequality known as less than or greater than.
The set of real numbers is an ordered set. Given any two numbers, we can determine
whether one number is less than, equal to, or greater than the other. Let’s see how this is
expressed symbolically.
We use the inequality symbol to represent “less than,” and we write
ab
This is read “a is less than b”
to indicate that a is less than b. The number line gives us a clear picture of the meaning of
this statement. The point corresponding to a must lie to the left of the point corresponding
to b.
a
b
ab
NOTE Notice that
a
b
and
ba
are equivalent statements. The
symbol “points to” the smaller
quantity.
Similarly, the inequality symbol represents “greater than,” and the statement
a
b
This is read “a is greater than b”
indicates that a is greater than b and means that the point corresponding to a on the number
line lies to the right of the point corresponding to b.
b
a
a
b
The following example illustrates the use of the inequality symbols.
Example 1
Establishing the Direction of Inequalities
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Complete each statement by inserting the symbol or between the given numbers.
(a) 2 ______ 8
28
(b) 2.786 ______ 2.78
2.786 2.78
(c) 23 ______ 5
23 5
(d) 12 ______ 1.4
Recall that 1.414 is an
12 1.4
approximation for 12.
CHECK YOURSELF 1
Insert an inequality symbol that makes each of the following a true statement.
(a) 5 ______ 2
(b) 3.14 ______ p
(c) 10 ______ 15
(d) 115 ______ 4
(e) 9.78 ______ 9.87
(f) 1.3 ______ 4
3
17
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CHAPTER 1
THE REAL NUMBERS
Suppose we are given an inequality of the form
x 1
The solution set for an inequality (as it is for an equation) is the set of all values for the
variable that make the inequality a true statement. A convenient way to picture that solution
set is by a graph on a number line. The following example illustrates.
Example 2
Graphing Inequalities
Graph the following set.
NOTE This set is read “the set
of all x such that x is less than 4.”
{xx 4}.
We want to include all real numbers less than 4, that is, to the left of 4 on the number line.
NOTE The parenthesis at 4
means that the point
corresponding to 4 is not
included in the graph.
Such a graph is called
an open half line.
0
4
CHECK YOURSELF 2
Graph the following sets.
(a) {xx 5}
(b) {xx 3}
Two other symbols, and , are also used in writing inequalities. In each case they
combine the inequality symbols for less than or greater than with the symbol for equality.
The following shows the use of these new symbols. The statement
NOTE This combines the
symbols < and and means
that either a < b or a b.
NOTE Here either a > b or
a b.
ab
is read “a is less than or equal to b.” Similarly,
ab
is read “a is greater than or equal to b.” We consider the graph of inequalities involving
these symbols in our next example.
Example 3
Graphing Inequalities
7
means that the point
2
7
corresponding to is included
2
in the graph. Such a graph is
called a closed half line.
Here we want all numbers to the right of
0
7
7
and including .
2
2
7
2
CHECK YOURSELF 3
Graph each of the following sets.
(a) {xx 7}
(b)
xx 4
3
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NOTE Here the bracket at
Graph of the following set.
7
xx 2
INEQUALITIES AND ABSOLUTE VALUE
NOTE The open circle is used
to indicate the open half line,
extending to the right, and not
including 3.
SECTION 1.3
19
Note: You may very well encounter a different notation for indicating the graphs of inequalities. This involves the use of circles to represent open and closed half lines. For
example, the graph of {xx 3} can be drawn as
0
3
and the graph of {xx 2} as
NOTE The closed circle is used
to indicate the closed half line,
extending to the left, and
including 2.
2
0
Our subsequent work with inequalities in Chapters 2, 4, 5 and 9 involves the use of a
double-inequality statement such as
3 x 4
This statement combines the two inequalities
NOTE The word “and” is
implied in any doubleinequality statement.
x 3
x4
and
or,
{xx 3 and x 4}
That is why it is sometimes called a compound inequality.
In our next example we look at the graphs of inequalities that have this form.
Example 4
Graphing Compound Inequalities
Graph of the following set.
{x3 x 4}
NOTE The parentheses indicate
that the endpoints, 3 and 4,
are not included in the graph.
This is called an open interval.
For the solution set of this double inequality, we want all points that lie to the right of
3 (x 3) and to the left of 4 (x 4). This means that we should include all points that
lie between 3 and 4.
3
0
4
CHECK YOURSELF 4
Graph each of the following sets.
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(a) {x1 x 6}
(b) {x2 x 8}
Once again, we refer to the number line to introduce our final topic of this section. If we
locate the number 4 and its additive inverse, 4, on the number line, we see that both numbers correspond to points that are the same distance (4 units) from the origin.
4 units
4
4 units
0
4
When we are concerned not with the direction (left or right) of a number from the origin,
but only with the distance from the origin, we refer to that number’s absolute value.
20
CHAPTER 1
THE REAL NUMBERS
An absolute value is the distance (on the number line) between the point named by that
real number and the origin. We indicate the absolute value of a number with vertical bars.
In general, we can define the absolute value of any real number a as
NOTE If a is negative, then its
additive inverse, a, must be
positive and we want a positive
absolute value. Say a 2, then
2 (2) 2.
a
if a is positive
if a is zero
if a is negative
a c 0
a
The use of the absolute value notation is illustrated in our final example.
Example 5
Evaluating Absolute Value Expressions
Evaluate each of the following expressions.
be 5.
(b) 2.5 2.5
(c) 12 12
(d) 12 12
(e) 5 5
(f) 3 7 3 7 10
CHECK YOURSELF 5
Evaluate each of the following expressions.
(a) 121
(b) 3.4
(c) 13 (d) 15
(e) 8
(f) 9 2
CHECK YOURSELF ANSWERS
1. (a) 5 2; (b) 3.14 p; (c) 10 15; (d) 115 4; (e) 9.78 9.87;
4
(f ) 1.3 3
2. (a) {xx 5}
(b) {xx 3}
0
5
(b) x x 3. (a) {xx 7}
0
43
7
4. (a) {x1 x 6}
1
3
0
4
3
0
0
(b) {x2 x 8}
6
2
0
8
(This is a half open interval.)
5. (a) 121; (b) 3.4; (c) 13; (d) 15; (e) 8; (f) 11
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NOTE 5 is 5, so 5 must
(a) 32 32
Name
1.3
Exercises
Section
Date
Insert an inequality symbol or an equal sign to make each of the following a true
statement.
ANSWERS
1. 8 ______ 3
2. 3 ______ 7
3. 6 ______ 4
4. 2 ______ 3
5. 2.6 ______ 3.8
6. 7.40 ______ 7.4
5
3
7. _______ 5
4
8. 1.2 ______ 4
3
10. 13 ______ 1.4
9. 12 ______ 1.4
7
11. 1.75 ______
4
8
12.
______ 2.33
3
13. 3 ______ 3
14. 3 ______ 1
15. 5 ______ 5
16. 4 ______ 4
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Write each of the following inequalities in words.
19.
17. y 2
18. x 3
19. m 3
20. n 5
21. a b
22. r s
23. b 0
24. y 0
25. 3 p 7
26. 4 y 2
20.
21.
22.
23.
24.
Graph each of the following sets. Assume x represents a real number.
27. {xx 4}
28.
0
{xx 5}
25.
0
26.
29. {xx 4}
30.
0
{xx 2}
0
27.
© 2001 McGraw-Hill Companies
28.
31. {x4 x}
32.
0
{x4 x}
29.
0
30.
33. {xx 4}
34.
0
{x0 x}
31.
0
32.
33.
34.
21
ANSWERS
35.
35. {x1 x 2}
36.
36.
0
37. {x4 x 1}
37.
38. {x2 x 5}
38.
{x3 x 5}
0
0
0
39.
Rewrite each of the following statements, using inequality symbols. Then graph the
solution set for each inequality. Assume that x represents a real number.
40.
39. x is less than 3
40. x is more than 2
41.
0
0
41. x is at least 1
42.
43.
42. x is no more than 5
0
43. x is greater than 4
44.
45.
0
44. x is at least 2
0
0
45. x is no more than 2
46.
47.
46. x is not less than 2
0
0
47. 2 is less than x, and x is less than 4
48.
0
49.
50.
51.
52.
48. 4 is less than or equal to x, and x is less than 2
0
Write each of the following expressions without the absolute value symbol.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
49. 5
50. 5
51. 2.5
52. 4.5
53. 2
52
6
72
8
55. 1.2
56. 4.5
57. 3
58. 6
59. 2
22
54. 2
22
3
7
2
60. 2 2
61. 3 5
62. 2 6
63. (3 4)
64. (4 8)
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53.
ANSWERS
65.
Label each statement as true or false. If it is false, explain.
66.
65. 2 2
66. 0 0
67. 12 12
68. 2 3 0 1
69. x x
67.
68.
Hint: For this statement to be true, it must be true for all values of the
variable.
69.
70. a a
71. The absolute value of any real number is positive or zero.
70.
72. Some real numbers have no absolute value.
71.
73. The absolute value of any real number is equal to the absolute value of its additive
inverse.
74. There is only one real number that is equal to its own absolute value.
72.
73.
75. Do you think that the following statement is true?
a b a b for all numbers a and b
When we don’t know whether such a statement is true, we refer to the statement as a
conjecture. We may “test” the conjecture by substituting specific numbers for the
variables.
Test the conjecture using two positive numbers for a and b.
Test again using a positive number for a and 0 for b.
Test again using two negative numbers.
Now try using one positive number and one negative number.
Summarize your results in a rule that you think is true.
74.
75.
76.
77.
In exercises 76 to 80, test the given conjecture, as you did in exercise 75.
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76. Do you think that the following statement is true?
78.
a b a b for all numbers a and b
77. Do you think that the following statement is true?
a b a b for all numbers a and b
78. Do you think that the following statement is true?
a b b a for all numbers a and b
23
ANSWERS
79. Do you think that the following statement is true?
79.
a
a
2 2 b
b
for all numbers a and b
80. Do you think that the following statement is true?
80.
a a for any number a
81. If a represents a positive number and b represents a negative number, determine
81.
whether the given expression is positive or negative.
(a) b a
(b) b (a)
(c) (b) a
(d) b a
Answers
1. 3. 5. 7. 9. 11. or 13. 15. 17. y is greater than or equal to 2
19. m is less than 3
21. a is greater than or equal to b
23. b is greater than 0
25. 3 is less than p and p is less than 7
x4
27.
0
31.
4
x 4
0
0
4
4 x 1
37.
1 0
39.
0
3
43.
0
x3
41.
x
4
45.
1 0
2
4
47.
0
x4
33.
1x2
0 1 2
55. 1.2
4
0
35.
4
x 4
29.
4
2
57. 3
4
2x4
59. 2
3
49. 5
61. 8
x 1
x 2
0
51. 2.5
63. 7
53.
5
6
65. True
81. (a) Positive; (b) negative; (c) positive; (d) positive
24
© 2001 McGraw-Hill Companies
67. True
69. False, it would not be true for negative values of x, for example, (1) 1.
71. True
73. True
75.
77.
79.