Solve. \( -5|x-4|>-20 \)

To solve the inequality: \[ -5|x - 4| > -20 \] **Step 1: Isolate the Absolute Value** First, divide both sides by \(-5\). Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign. \[ |x - 4| < 4 \] **Step 2: Remove the Absolute Value** The inequality \( |x - 4| < 4 \) means that the expression inside the absolute value is less than 4 units away from 0. This can be rewritten as a compound inequality: \[ -4 < x - 4 < 4 \] **Step 3: Solve for \( x \)** Add 4 to all parts of the inequality to isolate \( x \): \[ -4 + 4 < x < 4 + 4 \] \[ 0 < x < 8 \] **Step 4: Verify the Endpoints** Check the endpoints \( x = 0 \) and \( x = 8 \): - For \( x = 0 \): \[ -5|0 - 4| = -5 \times 4 = -20 \quad \text{(Not greater than -20)} \] - For \( x = 8 \): \[ -5|8 - 4| = -5 \times 4 = -20 \quad \text{(Not greater than -20)} \] Both endpoints do not satisfy the original inequality, so they are not included in the solution. **Final Solution:** All real numbers \( x \) between 0 and 8. \[ 0 < x < 8 \]