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. **Subtract 8 from both sides**:
\[
6c + 8 - 8 \leq -10 - 8
\]
\[
6c \leq -18
\]

2. **Divide both sides by 6**:
\[
c \leq \frac{-18}{6}
\]
\[
c \leq -3
\]

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

1. **Subtract 12 from both sides**:
\[
8c + 12 - 12 > 20 - 12
\]
\[
8c > 8
\]

2. **Divide both sides by 8**:
\[
c > \frac{8}{8}
\]
\[
c > 1
\]

### Combined Solution

Now we combine the solutions from both inequalities:

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

Thus, the solution to the compound inequality \( 6c + 8 \leq -10 \) or \( 8c + 12 > 20 \) is:

\[
c \leq -3 \quad \text{or} \quad c > 1
\]

This means any value of \( c \) that is less than or equal to \(-3\) or greater than \(1\) will satisfy the compound inequality.