Question

Solve for k.
9k+7≥18+10k or –17k–8+15k≤10
Write your answer as a compound inequality with integers.

Answer 1

To solve the compound inequality \( 9k + 7 \geq 18 + 10k \) or \( -17k - 8 + 15k \leq 10 \), we will solve each part separately.

### Part 1: Solve \( 9k + 7 \geq 18 + 10k \)

1. Start by subtracting \( 10k \) from both sides:
\[
9k - 10k + 7 \geq 18
\]
Simplifying gives:
\[
-k + 7 \geq 18
\]

2. Next, subtract 7 from both sides:
\[
-k \geq 11
\]

3. Multiply by -1 (note that this reverses the inequality):
\[
k \leq -11
\]

### Part 2: Solve \( -17k - 8 + 15k \leq 10 \)

1. Combine like terms on the left:
\[
(-17k + 15k) - 8 \leq 10
\]
This simplifies to:
\[
-2k - 8 \leq 10
\]

2. Add 8 to both sides:
\[
-2k \leq 18
\]

3. Divide by -2 (which reverses the inequality):
\[
k \geq -9
\]

### Combine results

Now we have two results:
- From the first part: \( k \leq -11 \)
- From the second part: \( k \geq -9 \)

Since the original inequality was an "or" situation, we combine these results:

The overall solution is:
\[
k \leq -11 \quad \text{or} \quad k \geq -9
\]

The final answer in compound inequality form is:
\[
(-\infty, -11] \cup [-9, \infty)
\]

If you want the integer solutions only, they would be:
- \( k \leq -11 \) includes integers \( \ldots, -13, -12, -11 \)
- \( k \geq -9 \) includes integers \( -9, -8, -7, \ldots \)

Therefore, the integer solution set is:
\[
k \in \{ \ldots, -13, -12, -11 \} \cup \{ -9, -8, -7, \ldots \}
\]