Let's find the solution for the inequality [tex]\(\frac{2}{3}c + 7 \leq \frac{1}{3}\)[/tex].
Step 1: Subtract 7 from both sides to isolate the term with [tex]\(c\)[/tex] on one side:
[tex]\[
\frac{2}{3}c + 7 - 7 \leq \frac{1}{3} - 7
\][/tex]
This simplifies to:
[tex]\[
\frac{2}{3}c \leq \frac{1}{3} - 7
\][/tex]
Step 2: Convert the numbers on the right side to have a common denominator for subtraction:
[tex]\[
\frac{1}{3} - 7 = \frac{1}{3} - \frac{21}{3} = \frac{1 - 21}{3} = \frac{-20}{3}
\][/tex]
So the inequality becomes:
[tex]\[
\frac{2}{3}c \leq \frac{-20}{3}
\][/tex]
Step 3: To solve for [tex]\(c\)[/tex], divide both sides by [tex]\(\frac{2}{3}\)[/tex]. Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[
\frac{2}{3}c \leq \frac{-20}{3}
\][/tex]
Multiplying both sides by [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[
c \leq \frac{-20 \cdot 3}{3 \cdot 2} = \frac{-60}{6} = -10
\][/tex]
Therefore, the solution to the inequality is:
[tex]\[
c \leq -10
\][/tex]
Selecting the correct multiple choice answer, we have:
[tex]\[
c \leq -10
\][/tex]
The correct answer is:
[tex]\[
\boxed{c \leq -10}
\][/tex]
