Problem 77 - College Algebra Essentials

Question

Solve each absolute value inequality. $$\left|3-\frac{2}{3} x\right|>5$$

Step-by-Step Solution

Answer

The solution to the absolute value inequality \(\left|3-\frac{2}{3} x\right|>5\) is \(x < -3\) or \(x > 12\).

Step 1: Divide the Inequality to Two Situations

The absolute value of an expression being greater than a number implies two possible situations. Write two inequalities to express these two possibilities: \(3-\frac{2}{3}x > 5\) and \(-1*(3-\frac{2}{3}x) > 5\).

Step 2: Solve The First Inequality

Solve the inequality to find the value of variable \(x\). Start by subtracting 3 on both sides, resulting in \(-\frac{2}{3}x > 2\). Next, multiply by \(-\frac{3}{2}\) to isolate \(x\). Note that flipping the inequality sign when multiplying by negative numbers results in \(x < -3\).

Step 3: Solve The Second Inequality

For the second inequality \(-1*(3-\frac{2}{3}*x) > 5\) or \(-3+\frac{2}{3}x > 5\), first add 3 to isolate the variable term. This results in \(\frac{2}{3}x > 8\). Multiply both side by \(\frac{3}{2}\) gives \(x > 12\).

Step 4: Summarize the Solution

This implies that the solution to this problem is that \(x\) is either less than -3 or greater than 12. These represent the two sets of possible values for \(x\) that satisfy the original inequality \(\left|3-\frac{2}{3} x\right|>5\).