We are given the inequality:

$- |2 - x| + 4 \geq -9$

Let's solve it step by step:

Step 1: Isolate the absolute value expression

Start by moving $4$ to the other side:

$- |2 - x| \geq -9 - 4$

$- |2 - x| \geq -13$

Now, divide both sides of the inequality by $-1$. Remember that dividing by a negative number reverses the inequality sign:

$|2 - x| \leq 13$

Step 2: Solve the absolute value inequality

The general solution for an absolute value inequality of the form $|A| \leq B$ is:

$-B \leq A \leq B$

So, we can rewrite the inequality:

$-13 \leq 2 - x \leq 13$

Step 3: Solve for $x$

Now, solve the inequality for $x$. First, subtract $2$ from all parts of the inequality:

$-13 - 2 \leq -x \leq 13 - 2$

$-15 \leq -x \leq 11$

Next, divide the inequality by $-1$, reversing the inequality signs:

$15 \geq x \geq -11$

This simplifies to:

$-11 \leq x \leq 15$

Final Answer:

The solution to the inequality is:

$x \in [-11, 15]$

This means $x$ can take any value between $-11$ and $15$, inclusive.

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Would you like further details or clarifications?

Here are 5 related questions for further practice:

1. Solve $|x + 3| - 2 \geq 4$.
2. Solve $|x - 5| \leq 7$.
3. Solve $2|x - 1| + 3 < 9$.
4. Solve $3 - |x - 4| > 0$.
5. Solve $|2x + 1| \geq 5$.

Tip: Always remember that dividing or multiplying an inequality by a negative number reverses the inequality sign.