A Simple Guide to Solving Absolute Value Inequalities

**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.

Solving Quadratic Inequalities: A Simple Step-by-Step Guide

A quadratic inequality involves a quadratic expression and requires finding the values of `x` that make the inequality true. Let’s break down the process in simple terms.

#### What is a Quadratic Inequality?

A **quadratic inequality** is an inequality that involves a quadratic expression and can be written in one of the following forms:

1. **Quadratic Less Than Zero:** `ax^2 + bx + c < 0`

2. **Quadratic Greater Than Zero:** `ax^2 + bx + c > 0`

3. **Quadratic Less Than or Equal To Zero:** `ax^2 + bx + c ≤ 0`

4. **Quadratic Greater Than or Equal To Zero:** `ax^2 + bx + c ≥ 0`

where `a`, `b`, and `c` are constants, and `a` is not zero.

#### Steps to Solve a Quadratic Inequality

Here’s a step-by-step guide:

1. **Rewrite the Inequality in Standard Form**

   Ensure the quadratic expression is in standard form, which is `ax^2 + bx + c` with the inequality sign.

2. **Find the Roots of the Quadratic Equation**

   Solve the corresponding quadratic equation `ax^2 + bx + c = 0` to find the roots. Use the quadratic formula:

   

   x = [-b ± sqrt(b^2 - 4ac)] / (2a)

   

   - **`b^2 - 4ac`** is called the discriminant.

   - **Positive Discriminant**: Two distinct real roots.

   - **Zero Discriminant**: One real root (a repeated root).

   - **Negative Discriminant**: No real roots.

3. **Plot the Roots on a Number Line**

   The roots divide the number line into intervals.

4. **Test Each Interval**

   Choose a test point from each interval and substitute it into the original inequality to see if the inequality holds.

5. **Write the Solution**

   Write the solution in interval notation, representing the ranges of `x` that satisfy the inequality.

#### Example: Solving a Quadratic Inequality

Solve the inequality:

x^2 - 3x - 4 < 0

**Step 1: Rewrite the Inequality**

   The inequality is in standard form.

**Step 2: Find the Roots**

   Solve `x^2 - 3x - 4 = 0` using the quadratic formula:

   

   x = [-(-3) ± sqrt((-3)^2 - 4*1*(-4))] / (2*1)

     = [3 ± sqrt(9 + 16)] / 2

     = [3 ± sqrt(25)] / 2

     = [3 ± 5] / 2

   

   So the roots are:

   

   x = (3 + 5) / 2 = 4

   x = (3 - 5) / 2 = -1

   

**Step 3: Plot the Roots**

   The roots are `x = -1` and `x = 4`. These points divide the number line into three intervals: `(-∞, -1)`, `(-1, 4)`, and `(4, ∞)`.

**Step 4: Test Each Interval**

   - For `x < -1` (e.g., `x = -2`):

     Substitute `x = -2` into `x^2 - 3x - 4`:

     

     (-2)^2 - 3*(-2) - 4 = 4 + 6 - 4 = 6 > 0

     

     This interval does not satisfy `x^2 - 3x - 4 < 0`.

   - For `-1 < x < 4` (e.g., `x = 0`):

     Substitute `x = 0` into `x^2 - 3x - 4`:

     

     0^2 - 3*0 - 4 = -4 < 0

     

     This interval satisfies `x^2 - 3x - 4 < 0`.

   - For `x > 4` (e.g., `x = 5`):

     Substitute `x = 5` into `x^2 - 3x - 4`:

     

     5^2 - 3*5 - 4 = 25 - 15 - 4 = 6 > 0

     

     This interval does not satisfy `x^2 - 3x - 4 < 0`.

**Step 5: Write the Solution**

   The solution to `x^2 - 3x - 4 < 0` is the interval where the inequality holds true:

   

   (-1, 4)

   

#### Summary

To solve a quadratic inequality:

1. Rewrite it in standard form.

2. Find the roots using the quadratic formula.

3. Plot the roots on a number line to create intervals.

4. Test each interval to see where the inequality holds.

5. Write the solution in interval notation.

Understanding these steps helps in solving quadratic inequalities and applying these concepts in various mathematical problems.