Select the correct answer.

Simplify the following expression.
[tex]\[ \left(\frac{1}{7} x+\frac{3}{8}\right)+\left(\frac{2}{9} x-\frac{1}{8}\right) \][/tex]

A. [tex]\(-\frac{5}{63} x+\frac{1}{2}\)[/tex]

B. [tex]\(\frac{3}{16} x+\frac{1}{4}\)[/tex]

C. [tex]\(\frac{23}{63} x+\frac{1}{2}\)[/tex]

D. [tex]\(\frac{23}{63} x+\frac{1}{4}\)[/tex]



Answer :

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]