To simplify the given expression:
[tex]\[
\left(\frac{1}{7} x+\frac{3}{8}\right)+\left(\frac{2}{9} x-\frac{1}{8}\right)
\][/tex]
we need to combine the like terms involving [tex]\( x \)[/tex] and the constant terms separately.
1. Combine the [tex]\( x \)[/tex]-terms:
[tex]\[
\frac{1}{7} x + \frac{2}{9} x
\][/tex]
To add these, we find a common denominator. The denominators are 7 and 9, so the least common multiple is 63. Rewrite each fraction with the common denominator:
[tex]\[
\frac{1}{7} x = \frac{1 \cdot 9}{7 \cdot 9} x = \frac{9}{63} x
\][/tex]
[tex]\[
\frac{2}{9} x = \frac{2 \cdot 7}{9 \cdot 7} x = \frac{14}{63} x
\][/tex]
Now, we can add the two fractions:
[tex]\[
\frac{9}{63} x + \frac{14}{63} x = \frac{9 + 14}{63} x = \frac{23}{63} x
\][/tex]
2. Combine the constant terms:
[tex]\[
\frac{3}{8} - \frac{1}{8}
\][/tex]
Both fractions already have the common denominator 8:
[tex]\[
\frac{3}{8} - \frac{1}{8} = \frac{3 - 1}{8} = \frac{2}{8} = \frac{1}{4}
\][/tex]
Putting it all together, the simplified expression is:
[tex]\[
\frac{23}{63} x + \frac{1}{4}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{23}{63} x + \frac{1}{4}}
\][/tex]
Which corresponds to option D:
D. [tex]\(\frac{23}{63} x + \frac{1}{4}\)[/tex]
