Let's go through the steps provided in the problem statement to determine the justification for Step 4.
We start with the given equation and simplify it through each step:
Step 1:
[tex]\[
\frac{9}{2}b + 11 - \frac{5}{6}b = b + 2
\][/tex]
We rewrite the equation on a common denominator:
[tex]\[
\frac{27}{6}b + 11 - \frac{5}{6}b = b + 2
\][/tex]
Combining the terms involving \( b \):
[tex]\[
\frac{22}{6}b + 11 = b + 2
\][/tex]
This is simplified to:
[tex]\[
\frac{11}{3}b + 11 = b + 2
\][/tex]
Step 2:
Subtract \( b \) from both sides to isolate the terms involving \( b \):
[tex]\[
\frac{8}{3}b + 11 = 2
\][/tex]
Step 3:
Subtract 11 from both sides to further isolate the \( b \) term:
[tex]\[
\frac{8}{3}b = -9
\][/tex]
Step 4:
To solve for \( b \), you need to isolate \( b \) by multiplying both sides of the equation by the reciprocal of \( \frac{8}{3} \), which is \( \frac{3}{8} \):
[tex]\[
b = -9 \times \frac{3}{8}
\][/tex]
[tex]\[
b = -\frac{27}{8}
\][/tex]
The operation performed in Step 4 is to multiply both sides of the equation by the reciprocal of \( \frac{8}{3} \), which is \( \frac{3}{8} \). This operation is known as the multiplication property of equality.
Thus, the correct justification for Step 4 is:
D. the multiplication property of equality
