https://file.aiquickdraw.com/mathbot/file/1725542910192wetloskb.jpg

The image shows the following compound inequality problem:

\[
3x - 1 < 14 \quad \text{and} \quad 4x + 5 > 11
\]

Let's solve each part of the compound inequality separately:

### 1. Solve \( 3x - 1 < 14 \):
\[
3x - 1 < 14
\]
Add 1 to both sides:
\[
3x < 15
\]
Divide both sides by 3:
\[
x < 5
\]

### 2. Solve \( 4x + 5 > 11 \):
\[
4x + 5 > 11
\]
Subtract 5 from both sides:
\[
4x > 6
\]
Divide both sides by 4:
\[
x > \frac{6}{4} = 1.5
\]

### 3. Combine the results:
The compound inequality requires both conditions to be satisfied simultaneously. Thus, the solution is:
\[
1.5 < x < 5
\]

This is the final solution, which can be graphed on the number line as an open interval from \(1.5\) to \(5\).

Do you need help graphing this, or would you like further details on any step?

Here are five related questions to further your understanding:
1. How would the solution change if the inequalities included "≤" or "≥" instead of "<" or ">"?
2. How do you solve a compound inequality if the word "or" is used instead of "and"?
3. Can compound inequalities have no solution? If so, when does that occur?
4. How would you graph \(1.5 < x < 5\) on a number line?
5. What happens if you multiply or divide an inequality by a negative number?

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

Solve the compound inequality: 3x - 1 < 14 and 4x + 5 > 11. Graph the solution on the number line.

Learn how to solve the compound inequality 3x - 1 < 14 and 4x + 5 > 11 with a step-by-step guide. This solution covers key algebraic concepts and demonstrates how to graph the solution on a number line. Suitable for students in Grades 8-10.

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