To solve the inequality \( |x + 5| \leq 7 \), we need to break it down into two cases based on the definition of absolute value.
- Case 1: \( x + 5 \leq 7 \)
- Case 2: \( -(x + 5) \leq 7 \) (which simplifies to \( x + 5 \geq -7 \))
Now we will solve each case separately.
For Case 1: \[ x + 5 \leq 7 \] Subtract 5 from both sides: \[ x \leq 2 \]
For Case 2: \[ -(x + 5) \leq 7 \] This can be rewritten as: \[ -x - 5 \leq 7 \] Adding 5 to both sides gives: \[ -x \leq 12 \] Multiplying by -1 (and reversing the inequality sign): \[ x \geq -12 \]
Combining the Results: From both cases, we have: \[ -12 \leq x \leq 2 \]
Therefore, the solution to the inequality \( |x + 5| \leq 7 \) is: \[ \boxed{[-12, 2]} \]