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Chapter 1: Problem 89

Solve each absolute value inequality. $$1<|2-3 x|$$

Short Answer

Expert verified
This absolute value inequality has no solution because the two ranges for x do not overlap.

Step by step solution

01

Break Down the Absolute Value Inequality

Break the inequality \(1<|2-3x|\) into two separate inequalities. This results in \(2-3x > 1\) and \(2-3x < -1\).
02

Solve the First Inequality

Solving \(2 - 3x > 1\) by first subtracting 2 from both sides to get \(-3x > -1\), then dividing both sides by -3 while flipping the inequality sign to get \(x < 1/3\).
03

Solve the Second Inequality

Solving \(2-3x < -1\) by first subtracting 2 from both sides to get \(-3x < -3\), then dividing both sides by -3 while flipping the inequality sign to get \(x > 1\).
04

Result

The x values that satisfy both inequalities are the solution to the absolute value inequality. Since there is no overlap between \(x < 1/3\) and \(x > 1\), there is no solution.

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