Problem 138 - College Algebra Essentials

Question

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

Step-by-Step Solution

Answer

The solution to the absolute value inequality |x - 2| < 3 is -1 < x < 5.

Step 1: Recognize the problem

Note that absolute value inequality |x| < a, where a is a positive number, means that all solutions x must lie between -a and a. It translates to -a < x < a in compound inequality form.

Step 2: Example and Solution

Consider an example of |x - 2| < 3. This translates to -3 < x - 2 < 3. Now, solve for x by adding 2 to all parts: => -1 < x < 5.

Step 3: Validate the Solution

Validate by substituting a value between -1 and 5 into the original inequality. For example, if x = 0, the left-hand side becomes |0 - 2| = 2 which is indeed less than 3. Thus, the solution is correct.