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Chapter 1: Problem 78

Solve each absolute value inequality. $$\left|3-\frac{3}{4} x\right|>9$$

Short Answer

Expert verified
The solutions to the inequality are \(x< -\frac{32}{3}\) or \(x > -16\).

Step by step solution

01

Split the Inequality

Because the absolute value is greater than 9, there are two cases to consider: when \(3-\frac{3}{4}x > 9\) and when \(3-\frac{3}{4}x < -9\). This splitting gives two inequalities to solve separately and will provide two ranges of solutions for \(x\).
02

Solve the First Inequality

Start by solving for the first inequality \(3-\frac{3}{4}x > 9\). Subtract 3 from both sides and then multiply by \(\frac{-4}{3}\). Remember, when multiplying an inequality by a negative number, the direction of the inequality changes: \(\frac{3}{4}x < -8\). This simplifies to \(x < - \frac{32}{3}\)
03

Solve the Second Inequality

Next, solve for the second inequality \(3-\frac{3}{4}x < -9\). Subtract 3 from both sides, then divide by -\(\frac{3}{4}\). The inequality becomes \(\frac{3}{4}x > -12\), which simplifies to \(x > -16\)
04

Combine the Solutions

We are looking for values that make the expression either greater than 9 or less than -9. Therefore, we must combine the ranges where the inequalities hold. The solution is \(x< -\frac{32}{3} \) or \( x > -16 \)

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