Solve the inequality.

To solve the compound inequality \(-3 \geq x + 2 \geq -4\), we can split it into two separate inequalities:

1. \(-3 \geq x + 2\)
2. \(x + 2 \geq -4\)

### Step 1: Solve the first inequality \(-3 \geq x + 2\)

Subtract 2 from both sides:

\[
-3 - 2 \geq x
\]
\[
-5 \geq x
\]
or
\[
x \leq -5
\]

### Step 2: Solve the second inequality \(x + 2 \geq -4\)

Subtract 2 from both sides:

\[
x \geq -4 - 2
\]
\[
x \geq -6
\]

### Step 3: Combine the results

So, we have:

- From \(x + 2 \geq -4\), we get \(x \geq -6\).
- From \(-3 \geq x + 2\), we get \(x \leq -5\).

This gives us the combined inequality:

\[
-6 \leq x \leq -5
\]

### Conclusion

The solution to the inequality is:

\[
[-6, -5]
\]

This means \(x\) can take any value from \(-6\) to \(-5\), inclusive.