https://file.aiquickdraw.com/mathbot/file/17264570246956uhl37ds.jpg

The compound inequality given in the image is:

\[
4u + 6 \leq -14 \quad \text{and} \quad 2u - 1 \geq 9
\]

We will solve each inequality separately and then find the intersection of the solutions.

### Step 1: Solve \( 4u + 6 \leq -14 \)
\[
4u + 6 \leq -14
\]
Subtract 6 from both sides:
\[
4u \leq -14 - 6
\]
\[
4u \leq -20
\]
Divide both sides by 4:
\[
u \leq -5
\]

### Step 2: Solve \( 2u - 1 \geq 9 \)
\[
2u - 1 \geq 9
\]
Add 1 to both sides:
\[
2u \geq 9 + 1
\]
\[
2u \geq 10
\]
Divide both sides by 2:
\[
u \geq 5
\]

### Step 3: Combine the solutions
From the first inequality, we have \( u \leq -5 \), and from the second inequality, we have \( u \geq 5 \).

There is no overlap between these two intervals, so there is no solution that satisfies both inequalities at the same time.

### Final Answer:
\[
\varnothing \quad \text{(no solution)}
\]

Would you like a detailed explanation of any step?

Here are 5 questions to expand your understanding:
1. How do we solve compound inequalities with "and" conditions?
2. What is interval notation, and how do we represent no solution in it?
3. What happens if there is no overlap between the solutions of two inequalities?
4. Can you explain why dividing by a positive number in inequalities doesn't reverse the inequality sign?
5. How would the solution change if the inequalities were joined by "or" instead of "and"?

**Tip:** When solving compound inequalities, always ensure to check for overlap when using "and" conditions.

Solve the compound inequality: 4u + 6 ≤ -14 and 2u - 1 ≥ 9. Write the solution in interval notation. If there is no solution, enter ∅.

Learn how to solve the compound linear inequality 4u + 6 ≤ -14 and 2u - 1 ≥ 9 with a step-by-step solution. This problem covers key algebraic concepts, such as solving inequalities and using interval notation. Suitable for students in Grades 8-10.

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