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Chapter 1: Problem 77

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

Short Answer

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

Step by step solution

01

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 \(-1*(3-\frac{2}{3}x) > 5\).
02

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 \(x < -3\).
03

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 \(x > 12\).
04

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 \(\left|3-\frac{2}{3} x\right|>5\).

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