Let's solve the given expression step by step:
The expression is:
[tex]\[
\frac{2x - 1}{4} + \frac{4x + 3}{6}
\][/tex]
To add these fractions, we need a common denominator. The denominators are 4 and 6. The least common multiple (LCM) of 4 and 6 is 12. So, we will convert both fractions to have a denominator of 12.
### Step 1: Convert [tex]\(\frac{2x - 1}{4}\)[/tex] to have a denominator of 12
Multiply both the numerator and the denominator by 3:
[tex]\[
\frac{2x - 1}{4} \times \frac{3}{3} = \frac{3(2x - 1)}{12} = \frac{6x - 3}{12}
\][/tex]
### Step 2: Convert [tex]\(\frac{4x + 3}{6}\)[/tex] to have a denominator of 12
Multiply both the numerator and the denominator by 2:
[tex]\[
\frac{4x + 3}{6} \times \frac{2}{2} = \frac{2(4x + 3)}{12} = \frac{8x + 6}{12}
\][/tex]
### Step 3: Add the two fractions
[tex]\[
\frac{6x - 3}{12} + \frac{8x + 6}{12} = \frac{ (6x - 3) + (8x + 6) }{12}
\][/tex]
Combine like terms in the numerator:
[tex]\[
(6x - 3) + (8x + 6) = 6x + 8x - 3 + 6 = 14x + 3
\][/tex]
So the combined fraction is:
[tex]\[
\frac{14x + 3}{12}
\][/tex]
Thus, the detailed computation results in the simplified fractional expression:
[tex]\[
\boxed{\frac{14x + 3}{12}}
\][/tex]
None of the provided options ([tex]\( \frac{14 x + 1}{12} \)[/tex], [tex]\( \frac{14 x + 2}{12} \)[/tex], nor [tex]\( \frac{14 x + 4}{12} \)[/tex]) is correct. The correct and simplified answer is [tex]\( \frac{14 x + 3}{12} \)[/tex]. None of the provided answer choices exactly match this correct result.
