Solving Absolute Value Equations and Inequalities
Chapter 2, Sections 6
- You must consider the absolute value equations and inequalities that follow to be forms.
That is, "x marks the spot."
In other words, "x" in the absolute value equations and inequalities below may be any algebraic expression.
The "a" is a real number.
- Geometric Interpretation of Absolute Value Equations and Inequalities
Consider a > 0...
Absolute
Value |
Solution |
Graphical
Solution |
| | x | = a |
x = -a and x = a |
 |
| | x | < a |
-a < x < a |
 |
| | x | > a |
x < -a or x > a |
 |
- Solving Absolute Value Equations and Inequalities
1) Consider a > 0...
- Note: x and y may be expressions.
Absolute
Value |
What
To Do |
Solution
Set |
| | x | = a |
Solve the 2 equations
x = -a and x = a |
{ x | x = -a or x = a } |
| | x | < a |
Solve the compound inequality
-a < x < a |
{ x | -a < x < a } |
| | x | > a |
Solve the 2 inequalities
x < -a and x > a |
{ x | x < -a or x > a } |
| | x | = | y | |
Solve the 2 inequalities
x = -y and x = y |
{ x | x = -y or x = y } |
| Note that the or means union |
2) Consider a < 0...
Remember that the absolute value of anything is always non-negative:
- | x |
 0, for any x.
Absolute
Value |
What
To Do |
Solution
Set |
| | x | = a |
Observe that there is no non-negative number (| x |) that can be negative (a < 0 is given). |
The empty set |
| | x | < a |
Observe that there is no non-negative number (| x |) that can be smaller than a negative number (a < 0 is given). |
The empty set |
| | x | > a |
Observe that anything non-negative (| x |) is always bigger that something negative (a < 0 is given). |
All Real numbers |
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