How to Solve Inequalities Involving Absolute Values: A Detailed Guide

When dealing with inequalities involving absolute values, it's crucial to approach them methodically. The solution process often requires breaking down the inequality into simpler cases. This guide will walk you through the detailed steps to solve the inequality |2x - 1| |1 - 2x|.

Understanding the Problem

The first step is to clearly understand the inequality: |2x - 1| |1 - 2x|. At first glance, this might appear to involve two variables (x1 and x). However, the problem revolves around a single variable, x, and the | | symbol, often referred to as the absolute value, denotes the distance from zero on the number line, regardless of direction.

Breaking Down the Inequality

Given the nature of absolute values, we need to consider different cases depending on the sign of the expressions inside the absolute value. This leads to splitting the inequality into two possible scenarios:

Case 1: (2x - 1) (1 - 2x)

First, subtract (1 - 2x) from both sides of the inequality:

2x - 1 1 - 2x

Then, add (2x) to both sides:

2x - 1 2x 1 - 2x 2x

This simplifies to:

4x - 1 1

Next, add 1 to both sides:

4x - 1 1 1 1

Which further simplifies to:

4x 2

Finally, divide both sides by 4:

x 2/4

This simplifies to:

x 1/2

Case 2: (2x - 1) -(1 - 2x)

First, add (1 - 2x) to both sides:

2x - 1 -(1 - 2x)

This simplifies to:

2x - 1 -1 2x

Next, subtract (2x) from both sides:

2x - 1 - 2x -1 2x - 2x

Which simplifies to:

-1 -1

We must reverse the sign of the inequality when multiplying or dividing by a negative number. Therefore:

-1 -1/3

This simplifies to:

x -2/3

Combining the Solutions

After solving both cases, we combine the solutions. The solutions to the inequality |2x - 1| |1 - 2x| are:

These are the intervals where the inequality holds true.

Visualization

To visualize the solution, you can use a graphing tool like Wolfram Alpha. Simply type |2x - 1| |1 - 2x| into the search bar for a graphical representation of the solution.

For more detailed insights and additional examples, consult with your instructor or refer to mathematical resources online.