To solve the equation [tex]\(\frac{x+1}{5} - 3 \cdot \frac{x-1}{10} = 2\)[/tex], follow these steps:
1. Combine the fractions: The common denominator for the fractions [tex]\(\frac{x+1}{5}\)[/tex] and [tex]\(\frac{3(x-1)}{10}\)[/tex] is 10. So, rewrite each fraction with this common denominator:
[tex]\[
\frac{x+1}{5} = \frac{2(x+1)}{10}
\][/tex]
[tex]\[
3 \cdot \frac{x-1}{10} = \frac{3(x-1)}{10}
\][/tex]
2. Rewrite the equation with a common denominator:
[tex]\[
\frac{2(x+1)}{10} - \frac{3(x-1)}{10} = 2
\][/tex]
3. Combine the fractions over the common denominator:
[tex]\[
\frac{2(x+1) - 3(x-1)}{10} = 2
\][/tex]
4. Simplify the numerator:
Distribute the 2 and -3 in the numerator:
[tex]\[
2(x+1) - 3(x-1) = 2x + 2 - 3x + 3 = -x + 5
\][/tex]
This gives us:
[tex]\[
\frac{-x + 5}{10} = 2
\][/tex]
5. Remove the fraction by multiplying both sides by 10:
[tex]\[
-x + 5 = 20
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
-x + 5 = 20 \implies -x = 20 - 5 \implies -x = 15 \implies x = -15
\][/tex]
Hence, the solution to the equation is [tex]\(x = -15\)[/tex].
Therefore, the correct answer is:
e. -15
