Problem 139 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: 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.