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.

---

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.

Learn how to solve the absolute value inequality −∣2−x∣+4≥−9 with a step-by-step explanation. This problem covers key algebraic concepts such as absolute value inequalities and reversing inequality signs when dividing by negative numbers. Suitable for students in Grades 8-10.

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