To solve the equation:
[tex]\[
\frac{x+4}{3}-\frac{x-4}{5}=2+\frac{3 x-1}{15}
\][/tex]
we will proceed step-by-step.
1. Identify the common denominator:
We see that the denominators are 3, 5, and 15. The least common multiple of these numbers is 15. Therefore, the common denominator is 15.
2. Rewrite each term with the common denominator of 15:
[tex]\[
\frac{x+4}{3} = \frac{5(x+4)}{15}
\][/tex]
[tex]\[
\frac{x-4}{5} = \frac{3(x-4)}{15}
\][/tex]
[tex]\[
2 = \frac{2 \cdot 15}{15} = \frac{30}{15}
\][/tex]
3. Rewrite the entire equation with the common denominator:
[tex]\[
\frac{5(x+4)}{15} - \frac{3(x-4)}{15} = \frac{30}{15} + \frac{3x-1}{15}
\][/tex]
4. Combine the fractions on both sides:
[tex]\[
\frac{5(x+4) - 3(x-4)}{15} = \frac{30 + (3x - 1)}{15}
\][/tex]
5. Expand and simplify the numerators:
[tex]\[
5(x+4) - 3(x-4) = 5x + 20 - 3x + 12 = 2x + 32
\][/tex]
[tex]\[
30 + 3x - 1 = 29 + 3x
\][/tex]
6. Combine the simplified fractions:
[tex]\[
\frac{2x + 32}{15} = \frac{29 + 3x}{15}
\][/tex]
7. Since the denominators are the same, set the numerators equal:
[tex]\[
2x + 32 = 29 + 3x
\][/tex]
8. Solve for [tex]\(x\)[/tex]:
[tex]\[
32 - 29 = 3x - 2x
\][/tex]
[tex]\[
3 = x
\][/tex]
Therefore, the solution to the equation is [tex]\( x = 3 \)[/tex].
So the correct answer is:
C) 3.
