https://file.aiquickdraw.com/mathbot/file/1728862214815gjbn73yk.jpg

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).

Resuelve la siguiente desigualdad y grafica: (x-2)/10 ≤ x + 2 ≤ 5/4

Learn how to solve the compound inequality (x-2)/10 ≤ x + 2 ≤ 5/4 step-by-step. This problem covers key algebraic concepts and demonstrates how to graph the solution on a number line. Suitable for students in Grades 9-11.

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