$$ 2\left\vert9-x\right\vert $$-7$$ \ge-1 $$

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?

---

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.

$$ 2\left\vert9-x\right\vert $$-7$$ \ge -1 $$

Learn how to solve the absolute value inequality 2|9 - x| - 7 ≥ -1 step by step. This example involves isolating the absolute value, applying key algebraic properties, and solving inequalities. Perfect for students in Grades 8-10.

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