Problem 89

Question

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

Step-by-Step Solution

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Answer

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

1Step 1: Break Down the Absolute Value Inequality

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

2Step 2: 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 $x < 1/3$.

3Step 3: 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 $x > 1$.

4Step 4: 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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