Solve the compound inequality.
4u+2<-14 and 3u+5≤17

Write the solution in interval notation.

We are asked to solve the compound inequality:

\[
4u + 2 < -14 \quad \text{and} \quad 3u + 5 \leq 17
\]

### Step 1: Solve the first inequality \( 4u + 2 < -14 \)

\[
4u + 2 < -14
\]
Subtract 2 from both sides:

\[
4u < -14 - 2
\]
\[
4u < -16
\]
Now, divide both sides by 4:

\[
u < -4
\]

### Step 2: Solve the second inequality \( 3u + 5 \leq 17 \)

\[
3u + 5 \leq 17
\]
Subtract 5 from both sides:

\[
3u \leq 17 - 5
\]
\[
3u \leq 12
\]
Now, divide both sides by 3:

\[
u \leq 4
\]

### Step 3: Combine the solutions

The solution to the first inequality is \( u < -4 \), and the solution to the second inequality is \( u \leq 4 \). Since this is a compound inequality with the word "and," we need to find the overlap (or intersection) of these two solutions.

The overlap occurs when \( u < -4 \), because this is more restrictive than \( u \leq 4 \).

### Step 4: Write the solution in interval notation

The solution is:
\[
(-\infty, -4)
\]

Would you like more details, or do you have any questions?

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Here are five related questions:
1. How would the solution change if the inequalities were combined with "or" instead of "and"?
2. What happens if we change the direction of one of the inequalities?
3. How can you graph this solution on a number line?
4. How would the solution differ if absolute values were involved in the inequality?
5. What if we needed to solve for \(u\) in a quadratic inequality instead?

**Tip:** Always pay attention to whether inequalities are combined with "and" or "or"—it affects the solution set.

Solve the compound inequality. 4u + 2 < -14 and 3u + 5 ≤ 17. Write the solution in interval notation.

Learn how to solve the compound inequality 4u + 2 < -14 and 3u + 5 ≤ 17 step-by-step. This solution covers core algebraic concepts, including solving inequalities and expressing the solution in interval notation. Suitable for Grades 8-10 students.

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