Question

Complete the following questions and show all of your work.

What are the solutions to the compound inequality 6c+8≤−10
or 8c+12>20
?

Answer 1

To solve the compound inequality \(6c + 8 \leq -10\) or \(8c + 12 > 20\), we will solve each inequality separately.

**First Inequality: \(6c + 8 \leq -10\)**

1. Start by isolating \(c\). We will first subtract 8 from both sides:
\[
6c + 8 - 8 \leq -10 - 8
\]
Simplifying gives:
\[
6c \leq -18
\]
2. Next, divide both sides by 6:
\[
c \leq -3
\]

**Second Inequality: \(8c + 12 > 20\)**

1. Isolate \(c\) by first subtracting 12 from both sides:
\[
8c + 12 - 12 > 20 - 12
\]
Simplifying gives:
\[
8c > 8
\]
2. Now, divide both sides by 8:
\[
c > 1
\]

**Combining Solutions:**

From the solutions obtained, we have:
- From the first inequality: \(c \leq -3\)
- From the second inequality: \(c > 1\)

Since this is a compound inequality with "or," the solution includes both parts.

Thus, the solutions to the compound inequality \(6c + 8 \leq -10\) or \(8c + 12 > 20\) are:
\[
c \leq -3 \quad \text{or} \quad c > 1
\]

**Final Answer:**
The solution set is \(c \leq -3\) or \(c > 1\).