How To Solve Inequalities With Absolute Values On Both Sides

How to Solve Inequalities with Absolute Values on Both Sides

Solving inequalities with absolute values on both sides can seem daunting, but with a systematic approach, you can master this skill. This guide breaks down the process, offering clear explanations and examples to help you confidently tackle these types of problems. Understanding how to solve these inequalities is crucial for various mathematical applications and problem-solving scenarios.

Understanding Absolute Value

Before diving into inequalities, let's refresh our understanding of absolute value. The absolute value of a number is its distance from zero on the number line. Therefore, it's always non-negative. For example, |3| = 3 and |-3| = 3.

Solving Inequalities with Absolute Values on Both Sides: A Step-by-Step Guide

The key to solving inequalities with absolute values on both sides lies in considering all possible cases. Here's a comprehensive guide:

1. Case 1: Both Expressions Inside the Absolute Value Are Non-Negative

• Scenario: This occurs when both expressions within the absolute value symbols are greater than or equal to zero.
• Approach: Simply remove the absolute value symbols and solve the resulting inequality as you would any other inequality. Remember to maintain the direction of the inequality sign.

Example: |x + 2| ≥ |x - 4|

Assume x + 2 ≥ 0 and x - 4 ≥ 0. This implies x ≥ -2 and x ≥ 4. Therefore, we consider x ≥ 4 for this case. Removing the absolute value symbols, we get:

x + 2 ≥ x - 4

Simplifying, we find 2 ≥ -4, which is always true. Therefore, the solution for this case is x ≥ 4.

2. Case 2: One Expression Inside the Absolute Value is Non-Negative, the Other is Negative

• Scenario: One expression inside the absolute value is greater than or equal to zero, and the other is less than zero.
• Approach: Remove the absolute value symbols, remembering to change the sign of the expression that was negative. Then solve the resulting inequality.

Example (Continuing from above):

Let's consider the case where x + 2 ≥ 0 (x ≥ -2) and x - 4 < 0 (x < 4). This means we're looking at the interval -2 ≤ x < 4.

Removing the absolute values gives us:

x + 2 ≥ -(x - 4)

Simplifying:

x + 2 ≥ -x + 4 2x ≥ 2 x ≥ 1

Combining this with our condition -2 ≤ x < 4, we get 1 ≤ x < 4.

3. Case 3: Both Expressions Inside the Absolute Value Are Negative

• Scenario: Both expressions inside the absolute value symbols are less than zero.
• Approach: Remove the absolute value symbols and change the signs of both expressions. Then, solve the resulting inequality.

Example (Continuing from above):

Now, consider the case where x + 2 < 0 (x < -2) and x - 4 < 0 (x < 4). This implies x < -2.

Removing absolute values:

-(x + 2) ≥ -(x - 4) -x - 2 ≥ -x + 4 -2 ≥ 4

This inequality is false, meaning there are no solutions in this case.

4. Combining the Solutions

The final solution is the union of the solutions from all cases. In our example, the solution to |x + 2| ≥ |x - 4| is the union of x ≥ 4 and 1 ≤ x < 4, which simplifies to x ≥ 1.

Advanced Considerations and Tips

• Squaring Both Sides: In some cases, squaring both sides of the inequality can simplify the problem. However, remember to carefully consider potential extraneous solutions introduced by squaring.
• Graphical Solutions: Visualizing the inequalities graphically can be very helpful, especially for more complex problems. Plot the functions represented by the expressions inside the absolute values and identify the regions where the inequality holds true.
• Practice Makes Perfect: The best way to master solving inequalities with absolute values is through consistent practice. Work through various examples, gradually increasing the complexity.

By understanding these steps and practicing regularly, you'll build the confidence and skills needed to solve even the most challenging inequalities involving absolute values on both sides. Remember to always check your solutions by substituting values back into the original inequality.