Let's solve the inequality step-by-step:
Given:
[tex]\[
10 \leq -\frac{2}{3}(9 + 12b)
\][/tex]
Step 1: Distribute [tex]\(-\frac{2}{3}\)[/tex] within the parentheses:
[tex]\[
-\frac{2}{3} \cdot 9 + (-\frac{2}{3}) \cdot 12b
\][/tex]
Calculate each term separately:
[tex]\[
-\frac{2}{3} \cdot 9 = -6
\][/tex]
[tex]\[
-\frac{2}{3} \cdot 12b = -8b
\][/tex]
Combining these, the inequality becomes:
[tex]\[
10 \leq -6 - 8b
\][/tex]
Step 2: Isolate the term with [tex]\(b\)[/tex]. Start by adding 6 to both sides:
[tex]\[
10 + 6 \leq -6 + 6 - 8b
\][/tex]
This simplifies to:
[tex]\[
16 \leq -8b
\][/tex]
Step 3: Solve for [tex]\(b\)[/tex]. Divide both sides by [tex]\(-8\)[/tex] and remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
\frac{16}{-8} \geq b
\][/tex]
[tex]\[
-2 \geq b
\][/tex]
or equivalently:
[tex]\[
b \leq -2
\][/tex]
Thus, the solution to the inequality is:
[tex]\[
\boxed{b \leq -2}
\][/tex]
The correct answer is:
A. [tex]\(b \leq -2\)[/tex]
