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Let's solve the given inequality step by step.

### The inequality system:
\[
4x - 2 \leq 8 \quad \text{and} \quad 8x - 1 > -9
\]

#### 1. Solve the first inequality \( 4x - 2 \leq 8 \):

\[
4x - 2 \leq 8
\]
Add 2 to both sides:
\[
4x \leq 10
\]
Divide both sides by 4:
\[
x \leq \frac{10}{4} = \frac{5}{2}
\]

So, from the first inequality:
\[
x \leq \frac{5}{2}
\]

#### 2. Solve the second inequality \( 8x - 1 > -9 \):

\[
8x - 1 > -9
\]
Add 1 to both sides:
\[
8x > -8
\]
Divide both sides by 8:
\[
x > -1
\]

So, from the second inequality:
\[
x > -1
\]

#### 3. Combine the two results:

We have:
\[
-1 < x \leq \frac{5}{2}
\]

This is the solution to the inequality.

### Answer:
- The correct choice from the options is:
  \[
  -1 < x \leq \frac{5}{2}
  \]
  
- The solution in interval notation is:
  \[
  (-1, \frac{5}{2}]
  \]

#### Graphing the solution:
- The solution is graphed on the number line from \(-1\) (open circle) to \(\frac{5}{2}\) (closed circle), including \(\frac{5}{2}\) but excluding \(-1\).

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Would you like more details, or have any questions about this solution?

Here are 5 related questions to expand your understanding:
1. How do we solve compound inequalities with "and"?
2. What happens if the inequalities involve "or" instead of "and"?
3. How do we solve inequalities with variables on both sides?
4. How can you represent the solution of an inequality on a number line?
5. What are the steps for solving quadratic inequalities?

**Tip:** When solving inequalities, remember that dividing or multiplying both sides by a negative number flips the inequality sign!

Solve the inequality 4x - 2 ≤ 8 and 8x - 1 > -9. Graph the solution on the number line and write the solution in interval notation.

Learn how to solve compound inequalities step by step, including solving 4x - 2 ≤ 8 and 8x - 1 > -9. Explore key concepts like linear inequalities and graph the solution on a number line. Suitable for students in Grades 9-10.

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