Let's solve the inequality step-by-step to find the value of [tex]\( b \)[/tex]:
The given inequality is:
[tex]\[
10 \leq -\frac{2}{3}(9 + 12b)
\][/tex]
Step 1: Distribute and simplify the expression inside the parentheses.
[tex]\[
-\frac{2}{3}(9 + 12b) = -\frac{2}{3} \cdot 9 + -\frac{2}{3} \cdot 12b
\][/tex]
[tex]\[
= -6 - 8b
\][/tex]
Thus, the inequality becomes:
[tex]\[
10 \leq -6 - 8b
\][/tex]
Step 2: Add 6 to both sides of the inequality to isolate the term with [tex]\( b \)[/tex]:
[tex]\[
10 + 6 \leq -8b
\][/tex]
[tex]\[
16 \leq -8b
\][/tex]
Step 3: Divide both sides by -8. Since we are dividing by a negative number, remember to reverse the inequality sign:
[tex]\[
\frac{16}{-8} \geq b
\][/tex]
[tex]\[
-2 \geq b
\][/tex]
Rewriting this, we get:
[tex]\[
b \leq -2
\][/tex]
So, the solution to the inequality is:
[tex]\[
b \leq -2
\][/tex]
Therefore, the correct answer is:
A. [tex]\( b \leq -2 \)[/tex]
