The equation in the image is:

$\left| \frac{x}{2} - 1 \right| = 2$

This is an absolute value equation. To solve it, recall that:

$|A| = B \implies A = B \, \text{or} \, A = -B \, \text{(if B ≥ 0)}.$

Here, $A = \frac{x}{2} - 1$ and $B = 2$.

Step 1: Split into two cases

$\frac{x}{2} - 1 = 2 \quad \text{or} \quad \frac{x}{2} - 1 = -2$

Step 2: Solve each case

Case 1: $\frac{x}{2} - 1 = 2$

1. Add $1$ to both sides: $\frac{x}{2} = 3$
2. Multiply through by $2$ to isolate $x$: $x = 6$

Case 2: $\frac{x}{2} - 1 = -2$

1. Add $1$ to both sides: $\frac{x}{2} = -1$
2. Multiply through by $2$ to isolate $x$: $x = -2$

Step 3: Verify the solutions

We substitute $x = 6$ and $x = -2$ back into the original equation:

1. For $x = 6$: $\left| \frac{6}{2} - 1 \right| = \left| 3 - 1 \right| = |2| = 2 \quad \text{(True)}$

2. For $x = -2$: $\left| \frac{-2}{2} - 1 \right| = \left| -1 - 1 \right| = |-2| = 2 \quad \text{(True)}$

Both solutions satisfy the original equation.

Final Solution:

The solutions are:

$\boxed{x = 6 \, \text{and} \, x = -2}$

Do you want me to explain any step in more detail? Let me know!

Here are 5 related questions you might find helpful:

1. How do you solve absolute value equations with more complex expressions?
2. What happens if an absolute value equation has no solution?
3. How do you graph an absolute value function?
4. How do you solve inequalities involving absolute values?
5. How does the absolute value equation change if there’s a coefficient outside the absolute value?

Tip: Always check both solutions when solving absolute value equations—they may not always satisfy the original equation!