∣x+4∣≤7

To solve this inequality, we can examine the cases when $x+4$ is positive and when $x+4$ is negative.

Case 1: $x+4 \geq 0$
This means that $x \geq -4$.
Substitute $x+4$ with $x+4$ in the original inequality:
$x+4 \leq 7$
$x \leq 3$

Case 2: $x+4 < 0$
This means that $x < -4$.
Since the absolute value of a negative number is positive, we can rewrite the original inequality as:
$-(x+4) \leq 7$
$-x-4 \leq 7$
$-x \leq 11$
$x \geq -11$

Combining the results from both cases, we get the solution:
$x \in [-11, 3]$