To solve the inequality [tex]\( \frac{3x - 1}{4} - \frac{x - 5}{5} > -2 \)[/tex], we will follow these steps:
1. Combine the fractions: Start by finding a common denominator for the fractions on the left side. The common denominator for 4 and 5 is 20.
[tex]\[
\frac{3x - 1}{4} = \frac{5(3x - 1)}{20} = \frac{15x - 5}{20}
\][/tex]
[tex]\[
\frac{x - 5}{5} = \frac{4(x - 5)}{20} = \frac{4x - 20}{20}
\][/tex]
So the inequality becomes:
[tex]\[
\frac{15x - 5}{20} - \frac{4x - 20}{20} > -2
\][/tex]
2. Combine the numerators: Subtract the fractions by combining the numerators over the common denominator:
[tex]\[
\frac{(15x - 5) - (4x - 20)}{20} > -2
\][/tex]
Simplifying inside the numerator:
[tex]\[
= \frac{15x - 5 - 4x + 20}{20} = \frac{11x + 15}{20} > -2
\][/tex]
3. Clear the fraction by multiplying both sides by 20: This eliminates the denominator.
[tex]\[
11x + 15 > -40
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Isolate [tex]\( x \)[/tex] on one side of the inequality.
[tex]\[
11x + 15 > -40 \implies 11x > -40 - 15 \implies 11x > -55 \implies x > \frac{-55}{11} \implies x > -5
\][/tex]
Therefore, the solution to the inequality is [tex]\( x > -5 \)[/tex].
Hence, among the given options, the correct answer is: [tex]\(\boxed{x > -5}\)[/tex].
