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Absolute value inequalities are mathematical expressions that involve the absolute value function and inequalities. The absolute value of a real number represents its distance from zero on the number line. Inequalities involving absolute values often require different approaches than regular inequalities due to the non-linear nature of the absolute value function.

Basic Concepts

To understand absolute value inequalities, it's crucial to grasp the concept of absolute value. For any real number x, the absolute value of x, denoted as |x|, is defined as:

|x| = x if x ≥ 0, and |x| = -x if x < 0.

When solving absolute value inequalities, we often encounter expressions in the form |ax + b| < c or |ax + b| > c, where a, b, and c are real numbers.

Solving Absolute Value Inequalities

To solve absolute value inequalities, we typically follow these steps:

1. Isolate the absolute value expression if it's not already isolated.
2. Set up two inequalities without absolute values by considering both positive and negative cases.
3. Solve each inequality separately.
4. Combine the solutions if needed and represent the final solution on the number line.

Examples

Let's consider a few examples to illustrate the process of solving absolute value inequalities:

Example 1:

Solve the inequality |2x - 3| < 5.

We start by isolating the absolute value expression:

|2x - 3| < 5

Then, we set up two inequalities:

-5 < 2x - 3 < 5

and

-5 < -2x + 3 < 5

We solve each inequality separately and combine the solutions to obtain the final solution.

Example 2:

Solve the inequality |3x + 2| >= 7.

We follow similar steps as in Example 1 to solve this absolute value inequality.

Conclusion

Absolute value inequalities are important in various fields of mathematics and real-life applications. Mastering the techniques to solve them is essential for a deeper understanding of algebra and related subjects.