Problem 116 - College Algebra

Question

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

Step-by-Step Solution

Answer

To solve an inequality involving '<', interpret it to mean that the value inside the absolute value sign must be less than the given number. Rewriting the inequality as a compound inequality will provide a solution set, expressed either as an inequality like \(-3 < x < 3\) or in interval notation as \((-3,3)\). These represent all real numbers between -3 and 3.

Step 1: Set Up the Inequality

The first step is to set up the inequality. For an inequality like \(|x| < a\), it represents that x is less than a units away from zero in both directions. As such, it can be rewritten as \(-a < x < a\). This structure will form the basis of solving the inequality.

Step 2: Solve for x

In the case of \(|x| < a\), rewrite it as \(-a < x < a\), which means that x is between -a and a. Solve these two inequalities separately, which usually doesn't need further steps, as they basically give you your solution.

Step 3: Write the Solution Set

The final step is to express the solution in set notation or interval notation. If a = 3, for example, the solution set will be \(-3 < x < 3\), which in interval notation is \((-3, 3)\). In this case, the solution indicates every real number between -3 and 3 is a solution to the inequality.