Solve the following inequality algebraically. \[ 4|x+4|+7 \geq 15 \]

To solve the inequality \( 4|x+4|+7 \geq 15 \), first, we isolate the absolute value term: 1. Subtract 7 from both sides: \[ 4|x+4| \geq 8 \] 2. Divide both sides by 4: \[ |x+4| \geq 2 \] Next, we solve the absolute value inequality. Recall that \( |A| \geq B \) translates to two cases: \( A \leq -B \) or \( A \geq B \). Thus, we have: - Case 1: \( x + 4 \leq -2 \) \[ x \leq -6 \] - Case 2: \( x + 4 \geq 2 \) \[ x \geq -2 \] Combining these results, the solution to the inequality is: \[ x \leq -6 \quad \text{or} \quad x \geq -2 \] In interval notation, the solution is: \[ (-\infty, -6] \cup [-2, \infty) \]