**Absolute value inequalities** involve expressions with absolute value symbols, such as `|x|`. The absolute value of a number is its distance from zero on a number line, regardless of direction. Here’s a straightforward guide to solving these types of inequalities.
#### What is an Absolute Value Inequality?
An **absolute value inequality** takes the form:
1. **Less Than Inequality:** `|x| < a`
2. **Greater Than Inequality:** `|x| > a`
3. **Less Than or Equal To Inequality:** `|x| ≤ a`
4. **Greater Than or Equal To Inequality:** `|x| ≥ a`
where `a` is a non-negative number (i.e., `a ≥ 0`).
#### Steps to Solve Absolute Value Inequalities
**1. Solve Absolute Value Less Than or Equal To Inequality**
For `|x| ≤ a`:
1. **Rewrite as Two Inequalities**
The inequality `|x| ≤ a` means the distance from `x` to 0 is at most `a`. So, it translates to:
-a ≤ x ≤ a
This gives you a range where `x` lies between `-a` and `a`.
2. **Solve the Compound Inequality**
Directly use the compound inequality to find the solution. For example, for `|x| ≤ 3`:
-3 ≤ x ≤ 3
The solution is all `x` in the interval `[-3, 3]`.
**2. Solve Absolute Value Greater Than or Equal To Inequality**
For `|x| ≥ a`:
1. **Rewrite as Two Separate Inequalities**
The inequality `|x| ≥ a` means the distance from `x` to 0 is at least `a`. So, it translates to:
x ≤ -a or x ≥ a
This means `x` is either less than or equal to `-a` or greater than or equal to `a`.
2. **Solve Each Inequality Separately**
For example, for `|x| ≥ 4`:
x ≤ -4 or x ≥ 4
The solution is all `x` in the intervals `(-∞, -4]` or `[4, ∞)`.
**3. Solving Absolute Value Less Than Inequality**
For `|x| < a`:
1. **Rewrite as Two Inequalities**
The inequality `|x| < a` means the distance from `x` to 0 is less than `a`. So, it translates to:
-a < x < a
This gives you a range where `x` lies between `-a` and `a`.
2. **Solve the Compound Inequality**
Directly use the compound inequality to find the solution. For example, for `|x| < 2`:
-2 < x < 2
The solution is all `x` in the interval `(-2, 2)`.
**4. Solving Absolute Value Greater Than Inequality**
For `|x| > a`:
1. **Rewrite as Two Separate Inequalities**
The inequality `|x| > a` means the distance from `x` to 0 is more than `a`. So, it translates to:
x < -a or x > a
This means `x` is either less than `-a` or greater than `a`.
2. **Solve Each Inequality Separately**
For example, for `|x| > 5`:
x < -5 or x > 5
The solution is all `x` in the intervals `(-∞, -5)` or `(5, ∞)`.
#### Examples
1. **Example 1: Solving |x| ≤ 4**
- Rewrite as two inequalities: `-4 ≤ x ≤ 4`
- Solution: The interval `[-4, 4]`
2. **Example 2: Solving |x| > 3**
- Rewrite as two inequalities: `x < -3 or x > 3`
- Solution: The intervals `(-∞, -3)` and `(3, ∞)`
#### Summary
To solve absolute value inequalities:
1. **For Less Than or Equal To (`|x| ≤ a`)**: Rewrite as `-a ≤ x ≤ a`.
2. **For Greater Than or Equal To (`|x| ≥ a`)**: Rewrite as `x ≤ -a or x ≥ a`.
3. **For Less Than (`|x| < a`)**: Rewrite as `-a < x < a`.
4. **For Greater Than (`|x| > a`)**: Rewrite as `x < -a or x > a`.
Understanding these steps helps in solving absolute value inequalities and applying these concepts to various mathematical problems.
