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.