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Chapter 0: Problem 72

Solve each absolute value inequality. $$|5 x-2|>13$$

Short Answer

Expert verified
The solution to the inequality \(|5x - 2| > 13\) is \(x < -2.2\) or \(x > 3\).

Step by step solution

01

Set up two separate inequalities

An absolute value inequality \(|a| > b\) can be split into two separate inequalities: \(a < -b\) or \(a > b\). So, the given inequality \(|5x - 2| > 13\) will become \(5x - 2 < -13\) or \(5x - 2 > 13\). These two cases correspond to the fact that any number that's farther away from 0 than 13 on the number line is either less than -13 or greater than 13.
02

Solve each inequality separately

For the first inequality, \(5x - 2 < -13\), add 2 to both sides to isolate \(5x\), giving \(5x < -11\). Then, divide both sides by 5 to solve for \(x\), getting \(x < -2.2\). For the second inequality, \(5x - 2 > 13\), once again add 2 to both sides to yield \(5x > 15\), and then divide by 5, yielding \(x > 3\).
03

Write the solution

The solutions to the original inequality are the solutions to these two separate inequalities. Since \(x\) has to satisfy the original inequality, the final solution is \(x < -2.2\) or \(x > 3\)

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