Question

Solve for c.
5≥ 4c+11/3 > –3
Write your answer as a compound inequality with integers.

Answer 1

To solve the compound inequality \(5 \geq \frac{4c + 11}{3} > -3\), we'll break it down into two parts and solve each part separately.

1. **Solving the left part**:
\[
5 \geq \frac{4c + 11}{3}
\]
Multiply both sides by 3 to eliminate the fraction (remember that multiplying by a positive number doesn't change the inequality):
\[
15 \geq 4c + 11
\]
Now, subtract 11 from both sides:
\[
15 - 11 \geq 4c
\]
\[
4 \geq 4c
\]
Now, divide both sides by 4:
\[
1 \geq c
\]
or
\[
c \leq 1
\]

2. **Solving the right part**:
\[
\frac{4c + 11}{3} > -3
\]
Again, multiply both sides by 3:
\[
4c + 11 > -9
\]
Subtract 11 from both sides:
\[
4c > -9 - 11
\]
\[
4c > -20
\]
Now, divide both sides by 4:
\[
c > -5
\]

Combining both results, the compound inequality is:
\[
-5 < c \leq 1
\]

This can be written as:
\[
c \in (-5, 1]
\]

In terms of integers, the integer values for \(c\) that satisfy this inequality are:
\[
c = -4, -3, -2, -1, 0, 1
\]

Putting this into a compound inequality with integers:
\[
-5 < c \leq 1, \text{ where } c \in \{-4, -3, -2, -1, 0, 1\}
\]

Thus, the final answer is:
\[
-5 < c \leq 1
\] with integer solutions as stated above.