To find the numerical length of the expression [tex]\(6x + 31\)[/tex] that matches one of the given choices, we need to determine the value of [tex]\(x\)[/tex] that would satisfy the condition [tex]\(6x + 31 = \text{one of the choices}\)[/tex].
Let's solve this step-by-step for each choice:
1. Choice a: 151
[tex]\[
6x + 31 = 151
\][/tex]
Subtract 31 from both sides:
[tex]\[
6x = 151 - 31
\][/tex]
[tex]\[
6x = 120
\][/tex]
Divide by 6:
[tex]\[
x = \frac{120}{6} = 20
\][/tex]
Therefore, for [tex]\( x = 20 \)[/tex], [tex]\( 6x + 31 = 151 \)[/tex].
2. Choice b: 115
[tex]\[
6x + 31 = 115
\][/tex]
Subtract 31 from both sides:
[tex]\[
6x = 115 - 31
\][/tex]
[tex]\[
6x = 84
\][/tex]
Divide by 6:
[tex]\[
x = \frac{84}{6} = 14
\][/tex]
Therefore, for [tex]\( x = 14 \)[/tex], [tex]\( 6x + 31 = 115 \)[/tex].
3. Choice c: 33
[tex]\[
6x + 31 = 33
\][/tex]
Subtract 31 from both sides:
[tex]\[
6x = 33 - 31
\][/tex]
[tex]\[
6x = 2
\][/tex]
Divide by 6:
[tex]\[
x = \frac{2}{6} \approx 0.333
\][/tex]
Therefore, for [tex]\( x \approx 0.333 \)[/tex], [tex]\( 6x + 31 \approx 33 \)[/tex].
4. Choice d: 14
[tex]\[
6x + 31 = 14
\][/tex]
Subtract 31 from both sides:
[tex]\[
6x = 14 - 31
\][/tex]
[tex]\[
6x = -17
\][/tex]
Divide by 6:
[tex]\[
x = \frac{-17}{6} \approx -2.833
\][/tex]
Therefore, for [tex]\( x \approx -2.833 \)[/tex], [tex]\( 6x + 31 \approx 14 \)[/tex].
After checking all choices, we see that the correct solution [tex]\( x = 20 \)[/tex] corresponds to choice (a) [tex]\( 151 \)[/tex]. Thus, the correct numerical length of [tex]\( 6x + 31 \)[/tex] is:
[tex]\[
\boxed{151}
\][/tex]
