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Chapter 1.8
Absolute Value and Inequalities
Recall from Chapter R that the absolute value of a
number a, written |a|, gives the distance from a to
0 on a number line. By this definition, the
equation |x| = 3 can be solved by finding all real
numbers at a distance of 3 units from 0.
|x| = 3
Two numbers satisfy this equation, 3 and -3, so the
solution set is {-3, 3}.
Distance
Distance
is 3.
Distance is is 3. Distance is less than 3.
Distance is
greater than 3.
greater than 3.
|
-3
|
0
|
3
Similarly |x| < 3 is satisfied by all real numbers
whose distances from 0 are less than 3, that is, the
interval -3<x<3 or (-3,3).
Finally x  3 is satisfied by all real numbers
whose distances from 0 are greater th an 3.
So the solution set is - ,-3  3,  .
Properties of Absolute Value
1. For b  0, a  b if and only if
a  b or a  - b
2. a  b if and only if
a  b or a  - b
3. a  b if and only if
-ba  b
4. a  b if and only if
a  b or a  b
Example 1 Solving Absolute Value Equations
Solve 5 - 3x  12
Example 1 Solving Absolute Value Equations
Solve 4x - 3  x  6
Example 2 Solving Absolute Value Inequalities
Solve 2x  1  7
Example 2 Solving Absolute Value Inequalities
Solve 2x  1  7
Example 3 Solving Absolute Value Inequalities Requiring a
Transformation
Solve 2 - 7x 1  4
Example 4 Solving Absolute Value Equations and Inequalities
Solve 2  5x  4
Example 4 Solving Absolute Value Equations and Inequalities
Solve 4x  7  3
Absolute Value Models for Distance and Tolerance
Recall from Selection R.2 that is a and b represent
two real numbers, then the absolute value of their
difference, either |a - b| or |b – a|, represent the
distance between them. This fact is used to write
absolute value equations or inequalities to express
distance.
Example 5 Using Absolute Value Inequaliteis to Describe Distances
Write each statement using an absolute value
inequality.
(a) k is no less than 5 units from 8
Example 5 Using Absolute Value Inequaliteis to Describe Distances
Write each statement using an absolute value
inequality.
(b) n is within .001 unit of 6
Example 6 Using Absolute Value to Model Tolerance
Suppose y = 2x + 1 and we want y to be within .01
unit of 4. For what values of x will this be true?
|y – 4| < .01
|2x + 1 – 4| < .01
Homework
Section 1.8 # 1 - 82