The inequality shown is:

$\frac{x - 2}{10} \leq x + 2 \leq \frac{5}{4}$

Step 1: Split the inequality into two parts

To solve it, let's break the compound inequality into two separate inequalities:

1. $\frac{x - 2}{10} \leq x + 2$
2. $x + 2 \leq \frac{5}{4}$

We'll solve these one by one.

Step 2: Solve the first inequality

$\frac{x - 2}{10} \leq x + 2$

Multiply both sides by 10 to eliminate the fraction:

$x - 2 \leq 10(x + 2)$

Expand:

$x - 2 \leq 10x + 20$

Subtract $x$ from both sides:

$-2 \leq 9x + 20$

Subtract 20 from both sides:

$-22 \leq 9x$

Divide by 9:

$x \geq -\frac{22}{9}$

Step 3: Solve the second inequality

Now, solve $x + 2 \leq \frac{5}{4}$:

$x \leq \frac{5}{4} - 2$

Convert 2 to quarters:

$x \leq \frac{5}{4} - \frac{8}{4} = -\frac{3}{4}$

Step 4: Combine the results

We have two inequalities:

$x \geq -\frac{22}{9} \quad \text{and} \quad x \leq -\frac{3}{4}$

Thus, the solution is:

$-\frac{22}{9} \leq x \leq -\frac{3}{4}$

Step 5: Graphing the solution

On a number line, the interval from $-\frac{22}{9}$ to $-\frac{3}{4}$ will be shaded, with closed dots at both endpoints since the inequality includes the equal sign ($\leq$).

Would you like a graphical representation of this? Let me know if you'd like more details on any step!

Follow-up Questions:

1. How would you handle a situation where there are more than two inequalities combined?
2. What happens if we change the inequality signs to strict inequalities ($<$)?
3. Can you describe how to graph inequalities that do not include equal signs?
4. What is the significance of multiplying both sides of an inequality by a negative number?
5. How do you solve a system of inequalities involving both $>$ and $\geq$ symbols?

Tip:

When solving inequalities involving fractions, always check if multiplying by a constant requires flipping the inequality sign (only happens if multiplying by a negative number).