We are given the inequality:

$2 |9 - x| - 7 \geq -1$

Step 1: Isolate the absolute value term.

Add $7$ to both sides of the inequality:

$2 |9 - x| \geq -1 + 7$

Simplifying:

$2 |9 - x| \geq 6$

Step 2: Divide by 2.

$|9 - x| \geq 3$

Step 3: Solve the absolute value inequality.

The general rule for absolute value inequalities is:

$|A| \geq B \quad \text{means} \quad A \leq -B \quad \text{or} \quad A \geq B$

So, we can write:

$9 - x \leq -3 \quad \text{or} \quad 9 - x \geq 3$

Step 4: Solve each inequality.

For $9 - x \leq -3$:

$-x \leq -12 \quad \text{(Subtract 9 from both sides)}$ $x \geq 12 \quad \text{(Divide both sides by -1, flip the inequality)}$

For $9 - x \geq 3$:

$-x \geq -6 \quad \text{(Subtract 9 from both sides)}$ $x \leq 6 \quad \text{(Divide both sides by -1, flip the inequality)}$

Step 5: Combine the solutions.

The solution is:

$x \geq 12 \quad \text{or} \quad x \leq 6$

In interval notation, the solution is:

$(-\infty, 6] \cup [12, \infty)$

This means that $x$ is less than or equal to 6, or greater than or equal to 12.

Would you like more details or have any questions?

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Here are 5 related questions for practice:

1. How would the solution change if the inequality was $2 |9 - x| - 7 > -1$?
2. Can you solve $|5 - x| \leq 4$ using the same approach?
3. How would you solve $-3 |x - 2| + 4 \geq 1$?
4. How do you graph the solution $(-\infty, 6] \cup [12, \infty)$ on a number line?
5. What happens if the absolute value term is multiplied by a negative number, like $-2 |9 - x| - 7 \geq -1$?

Tip: When solving absolute value inequalities, remember that you need to consider two cases: one where the expression inside the absolute value is positive and one where it's negative.