To solve the problem of adding the rational expressions \(\frac{3x + 6}{24}\) and \(\frac{2x - 1}{8}\), follow these steps:
1. Identify the least common denominator (LCD):
- The denominators of the given fractions are 24 and 8.
- The least common denominator of 24 and 8 is 24.
2. Rewrite each fraction with the common denominator:
- The fraction \(\frac{3x + 6}{24}\) already has 24 as the denominator.
- To rewrite \(\frac{2x - 1}{8}\) with 24 as the denominator, express it in terms of 24:
[tex]\[
\frac{2x - 1}{8} = \frac{2x - 1}{8} \times \frac{3}{3} = \frac{3(2x - 1)}{24} = \frac{6x - 3}{24}
\][/tex]
3. Add the fractions:
- Now, add the two fractions:
[tex]\[
\frac{3x + 6}{24} + \frac{6x - 3}{24} = \frac{(3x + 6) + (6x - 3)}{24} = \frac{3x + 6 + 6x - 3}{24} = \frac{9x + 3}{24}
\][/tex]
4. Simplify the result:
- Simplify the numerator if possible:
[tex]\[
\frac{9x + 3}{24} = \frac{3(3x + 1)}{24}
\][/tex]
- Recognize that the greatest common divisor (GCD) of 3 and 24 is 3, so we divide both the numerator and the denominator by 3:
[tex]\[
\frac{3(3x + 1)}{24} = \frac{3x + 1}{8}
\][/tex]
Therefore, the simplified form of the sum [tex]\(\frac{3x + 6}{24} + \frac{2x - 1}{8}\)[/tex] is [tex]\(\boxed{\frac{3x + 1}{8}}\)[/tex].
