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The given compound inequality is:

\[
-1 < 3x - 4 \leq 14
\]

### Step-by-step solution:

We will solve the compound inequality in two parts. This is essentially solving two inequalities combined into one.

#### Part 1: Solve \(-1 < 3x - 4\)

1. Add \(4\) to both sides:
   \[
   -1 + 4 < 3x
   \]
   \[
   3 < 3x
   \]

2. Divide both sides by \(3\):
   \[
   1 < x
   \]
   \[
   x > 1
   \]

#### Part 2: Solve \(3x - 4 \leq 14\)

1. Add \(4\) to both sides:
   \[
   3x - 4 + 4 \leq 14 + 4
   \]
   \[
   3x \leq 18
   \]

2. Divide both sides by \(3\):
   \[
   x \leq 6
   \]

### Combined Solution:

We now combine the two parts:
\[
1 < x \leq 6
\]
This means the solution for \(x\) is all values between \(1\) and \(6\), but \(x\) is strictly greater than \(1\) and less than or equal to \(6\).

### Graph on the number line:

On the number line:
- There will be an open circle at \(x = 1\) (because \(x > 1\), not including \(1\)).
- There will be a closed circle at \(x = 6\) (because \(x \leq 6\), including \(6\)).
- A line segment will connect these two points.

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Do you have any questions or would you like further details on this solution? Here are 5 related questions to expand your understanding:

1. How would you solve the compound inequality \( -3 \leq 2x - 5 < 7 \)?
2. What does an open circle and closed circle on the number line represent in inequalities?
3. How do you graph the solution of \(x > 2\) and \(x \leq 8\) on a number line?
4. What is the difference between solving a compound inequality and a simple inequality?
5. Can compound inequalities have more than two parts?

**Tip:** Always check if the inequality signs are strict (< or >) or inclusive (≤ or ≥) before graphing your solution!

Solve the compound inequality: -1 < 3x - 4 ≤ 14. Graph the solution on the number line.

Learn how to solve the compound inequality -1 < 3x - 4 ≤ 14 with a step-by-step solution. This problem involves key algebraic concepts, including manipulating and solving compound inequalities, and graphing the solution on the number line. Suitable for students in Grades 8-10.

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