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Let's solve the given compound inequality step by step:

The inequality is:

\[
-7 < 2x - 9 \leq 1
\]

### Step 1: Solve the inequality in parts.

#### First part: 
\[
-7 < 2x - 9
\]

Add 9 to both sides:

\[
-7 + 9 < 2x
\]
\[
2 < 2x
\]

Now, divide both sides by 2:

\[
1 < x
\]

#### Second part:
\[
2x - 9 \leq 1
\]

Add 9 to both sides:

\[
2x \leq 1 + 9
\]
\[
2x \leq 10
\]

Now, divide both sides by 2:

\[
x \leq 5
\]

### Step 2: Combine the results.

From the two parts, we have:

\[
1 < x \leq 5
\]

### Step 3: Write the solution in interval notation.

Since \(1 < x \leq 5\), the interval notation is:

\[
(1, 5]
\]

### Step 4: Graph the solution on the number line.

- The interval is from 1 to 5.
- At 1, use an open circle since \(x > 1\).
- At 5, use a closed circle since \(x \leq 5\).

### Answer:
- The correct multiple-choice answer is: **\(1 < x \leq 5\)**.
- The interval notation is: **(1, 5]**.

Let me know if you'd like further clarification on any of these steps!

---

Here are some related questions you might want to explore:

1. How do you solve compound inequalities with different inequality symbols?
2. What is the difference between an open and closed interval on a number line?
3. Can a compound inequality have no solution? In what cases?
4. How do you graph inequalities on a number line with infinity in the solution?
5. What is interval notation, and how does it differ from set-builder notation?

**Tip:** Always remember to handle each inequality in a compound inequality separately before combining the results!

Learn how to solve the compound inequality -7 < 2x - 9 ≤ 1 with step-by-step instructions. This example uses basic inequality principles and is perfect for students in Grades 8-10.

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