Problem 76 - College Algebra Essentials

Question

Solve each absolute value inequality. $$\left|\frac{3 x-3}{9}\right| \geq 1$$

Step-by-Step Solution

Answer

The solutions to the inequality are \( x \geq \frac{4}{3} \) and \( x \leq \frac{2}{3} \)

Step 1: Simplify the absolute value inequality

The equation inside the absolute value can be simplified. In order to simplify, divide both sides of the equation by 3. Therefore, the inequality is \( |x - 1| \geq \frac{1}{3} \)

Step 2: Remove the absolute value

When we remove the absolute value, we need to consider both negative and positive possibilities and make two equations accordingly. For the positive case, the equation remains the same \( x - 1 \geq \frac{1}{3} \). For the negative case, we flip the inequality and the sign of the expression inside the absolute value, so the negative equation is \( x - 1 \leq -\frac{1}{3} \)

Step 3: Solve for x

Solve the two inequalities. Add 1 to both sides of both equations to isolate x. The solution for the positive case is \( x \geq \frac{4}{3} \) and the solution for the negative case is \( x \leq \frac{2}{3} \)