22. \( -3\left|1-\frac{2}{3} v\right|=-9 \)

To solve the equation: \[ -3\left|1 - \frac{2}{3}v\right| = -9 \] **Step 1: Isolate the Absolute Value** Divide both sides by \(-3\) to isolate the absolute value: \[ \left|1 - \frac{2}{3}v\right| = \frac{-9}{-3} \implies \left|1 - \frac{2}{3}v\right| = 3 \] **Step 2: Remove the Absolute Value** An equation of the form \(|A| = B\) has two solutions: \(A = B\) or \(A = -B\). Apply this to the equation: 1. **First Case:** \[ 1 - \frac{2}{3}v = 3 \] Subtract 1 from both sides: \[ -\frac{2}{3}v = 2 \] Multiply both sides by \(-\frac{3}{2}\): \[ v = -3 \] 2. **Second Case:** \[ 1 - \frac{2}{3}v = -3 \] Subtract 1 from both sides: \[ -\frac{2}{3}v = -4 \] Multiply both sides by \(-\frac{3}{2}\): \[ v = 6 \] **Solution:** The values of \(v\) that satisfy the equation are: \[ v = -3 \quad \text{and} \quad v = 6 \] **Verification:** To ensure these solutions are correct, substitute them back into the original equation. 1. **For \(v = -3\):** \[ -3\left|1 - \frac{2}{3}(-3)\right| = -3\left|1 + 2\right| = -3(3) = -9 \] Which matches the right side. 2. **For \(v = 6\):** \[ -3\left|1 - \frac{2}{3}(6)\right| = -3\left|1 - 4\right| = -3(3) = -9 \] Which also matches the right side. Both solutions are valid. **Final Answer:** \[ v = -3 \quad \text{and} \quad v = 6 \]