Problem 69 - College Algebra Essentials

Question

Solve each absolute value inequality. $$|x|>3$$

Step-by-Step Solution

Answer

The solutions to the inequality \(|x| > 3\) are \(x > 3\) and \(x < -3\).

Step 1: Understand the Problem

The equations says that the absolute value of 'x' is greater than 3. This means that 'x' is either greater than 3 or less than -3. This concept can be clearly explained through a number line.

Step 2: Break into Two Inequalities

The inequality \(|x| > 3\) can be broken down into two inequalities: \(x > 3\) and \(x < -3\). This is because absolute value of 'x' gives positive values, so we're looking for values of 'x' that are more than 3 units away from 0 on either side.

Step 3: Solve Both Inequalities

Solving both \(x > 3\) and \(x < -3\) will give the solution to this problem. For \(x > 3\), any value larger than '3' would satisfy the equation. Similarly, for \(x < -3\), any value lesser than '-3' would satisfy the equation.

Step 4: Represent on a Number Line

Both of the solutions can be represented on a number line. Any point on the right side of '3' or on the left side of '-3' would satisfy the original inequality \(|x| > 3\).