To determine the value of [tex]\(b\)[/tex] given the function [tex]\( g(x) = \frac{1}{3x - 7} \)[/tex] and the equation [tex]\( g(b) = 20 \)[/tex], we proceed as follows:
1. Start by setting up the equation:
[tex]\[
g(b) = 20
\][/tex]
Substituting the definition of [tex]\( g(x) \)[/tex] into this equation gives:
[tex]\[
\frac{1}{3b - 7} = 20
\][/tex]
2. Solve for [tex]\( b \)[/tex]:
First, take the reciprocal of both sides to get rid of the fraction:
[tex]\[
3b - 7 = \frac{1}{20}
\][/tex]
3. Isolate [tex]\( b \)[/tex]:
Add 7 to both sides of the equation:
[tex]\[
3b = \frac{1}{20} + 7
\][/tex]
To combine terms, rewrite [tex]\( 7 \)[/tex] as a fraction with a common denominator:
[tex]\[
7 = \frac{140}{20}
\][/tex]
Then add:
[tex]\[
3b = \frac{1}{20} + \frac{140}{20} = \frac{141}{20}
\][/tex]
4. Solve for [tex]\( b \)[/tex] by dividing both sides by 3:
[tex]\[
b = \frac{\frac{141}{20}}{3} = \frac{141}{20} \times \frac{1}{3} = \frac{141}{60}
\][/tex]
Thus, the value of [tex]\( b \)[/tex] is:
[tex]\[
\boxed{\frac{141}{60}}
\][/tex]
Therefore, the correct answer is (B) [tex]\( \frac{141}{60} \)[/tex].
