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Chapter 1: Problem 139

Describe how to solve an absolute value inequality involving the symbol \(>.\) Give an example.

Short Answer

Expert verified
To solve an absolute value inequality |x| > a, split it into two inequalities: x > a or x < -a. Solve each inequality separately to determine the solution set for the original inequality.

Step by step solution

01

Understand the problem

The most critical step in any math problem is to understand what the problem asks. For an absolute value inequality involving the symbol '>', the task is to find all values of a variable where the absolute value of the variable is greater than a given number.
02

Set Up Two Inequalities

An absolute value inequality with '>' splits into two separate inequalities. If the inequality is |x| > a, the two inequalities are x > a and x < -a.
03

Solve Each Inequality Separately

After you have split the absolute value inequality into two separate inequalities, solve each inequality separately by isolating x on one side of the inequality.
04

Write down the solution set

The solution set of the inequality |x| > a is the set of all real numbers that satisfy either of the inequalities x > a or x < -a. This can be represented on a number line to visualize the solutions.
05

Example

Suppose we have to solve the inequality |x| > 3. This inequality becomes two separate inequalities: x > 3 or x < -3. So any number greater than 3 or less than -3 will be a solution to the inequality. This can be easily represented on the number line with a solid dot on -3 and 3 and shading to the left of -3 and to the right of 3 to depict all possible solutions.

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