solve the inequality. Write your answer using interval notation. |2 − x| − 4 ≥ −3

The inequality to solve is:

\[
|2 - x| - 4 \geq -3
\]

### Step 1: Isolate the absolute value
Add \(4\) to both sides to isolate the absolute value expression:

\[
|2 - x| \geq -3 + 4
\]
\[
|2 - x| \geq 1
\]

### Step 2: Solve the absolute value inequality
Now, we solve the inequality \( |2 - x| \geq 1 \).

Recall that the absolute value inequality \( |A| \geq B \) can be rewritten as two inequalities:
\[
A \geq B \quad \text{or} \quad A \leq -B
\]
So, for our case:
\[
2 - x \geq 1 \quad \text{or} \quad 2 - x \leq -1
\]

### Step 3: Solve each inequality

#### 1. \(2 - x \geq 1\)
Subtract 2 from both sides:
\[
-x \geq -1
\]
Multiply by \(-1\) and reverse the inequality sign:
\[
x \leq 1
\]

#### 2. \(2 - x \leq -1\)
Subtract 2 from both sides:
\[
-x \leq -3
\]
Multiply by \(-1\) and reverse the inequality sign:
\[
x \geq 3
\]

### Step 4: Combine the solutions
The solution is \(x \leq 1\) or \(x \geq 3\). In interval notation, this is:

\[
(-\infty, 1] \cup [3, \infty)
\]

### Final Answer:
\[
x \in (-\infty, 1] \cup [3, \infty)
\]

Would you like more details on any step or further questions? Here are some related questions for further practice:

1. How do you solve inequalities involving two absolute values?
2. What is the graphical interpretation of absolute value inequalities?
3. How do you approach solving \( |x + a| \leq b \) for \(a\) and \(b\)?
4. Can absolute value inequalities ever have no solution?
5. What are the key differences between solving \( |x| \geq b \) and \( |x| \leq b \)?

**Tip:** When solving absolute value inequalities, remember to break them into two cases, reflecting the positive and negative possibilities of the expression inside the absolute value.

Learn how to solve the absolute value inequality |2 − x| − 4 ≥ −3 using interval notation. This step-by-step solution covers key algebraic concepts such as absolute value inequalities and properties of inequalities, suitable for students in Grades 9-11.

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