To add the fractions [tex]\(\frac{5}{x}\)[/tex] and [tex]\(\frac{7}{x-4}\)[/tex], we need to find a common denominator. The denominators are [tex]\(x\)[/tex] and [tex]\(x-4\)[/tex]. The common denominator will be the product of these two denominators: [tex]\(x \cdot (x - 4) = x(x - 4)\)[/tex].
Next, we express each fraction with this common denominator:
[tex]\[
\frac{5}{x} = \frac{5(x-4)}{x(x-4)}
\][/tex]
[tex]\[
\frac{7}{x-4} = \frac{7x}{x(x-4)}
\][/tex]
Now add the two fractions:
[tex]\[
\frac{5(x-4)}{x(x-4)} + \frac{7x}{x(x-4)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{5(x-4) + 7x}{x(x-4)}
\][/tex]
Distribute and simplify the numerator:
[tex]\[
\frac{5x - 20 + 7x}{x(x-4)} = \frac{12x - 20}{x(x-4)}
\][/tex]
We can further simplify the numerator by factoring out the common factor of 4:
[tex]\[
\frac{4(3x - 5)}{x(x-4)}
\][/tex]
So the simplified form of [tex]\(\frac{5}{x} + \frac{7}{x-4}\)[/tex] is:
[tex]\[
\boxed{\frac{4(3x - 5)}{x(x - 4)}}
\][/tex]
