Problem 78 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(x\).

Step 2: 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 \(x < - \frac{32}{3}\)

Step 3: 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 \(x > -16\)

Step 4: 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 \( x > -16 \)