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Chapter 0: Problem 75

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

Short Answer

Expert verified
The solution for the inequality \(|3-(2/3)x|>5\) is \(x \in (-\infty,-3) \cup (12, \infty)\)

Step by step solution

01

Isolate the Absolute Value

The absolute value is already isolated in the inequality \(|3-(2/3)x|>5\). So, we proceed to the next step.
02

Break down the inequality

Since the absolute value is greater than a number, it means the expression inside the absolute value can be greater than that number or less than the negative of that number. Thus, the inequality \(|3-(2/3)x|>5\) leads to two inequalities: \((3-(2/3)x)>5\) and \((3-(2/3)x)<-5\)
03

Solve each inequality separately

1. Solve \((3-(2/3)x)>5\): Here we first take 3 to the other side: \(-(2/3)x > 5 - 3\). So, \(-(2/3)x > 2\). Now divide both sides by -2/3 (note that the inequality sign flips when you divide or multiply with negative number), we find: \(x < -3\). 2. Solve \((3-(2/3)x)<-5\): Here as well we first take 3 to the other side: \(-(2/3)x < -5 - 3\). So, \(-(2/3)x < -8\). Dividing both sides by -2/3 again gives us: \(x > 12\).
04

Combine the solution sets

From Step 3, we found \(x < -3\) and \(x > 12\). The solution to original inequality is the union of these two solution sets. So, \(x \in (-\infty,-3) \cup (12, \infty)\), which means x is either less than -3 or greater than 12.

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