To solve the equation [tex]\(\frac{x}{x-4}-\frac{8}{4-x}=2\)[/tex], let's follow these steps:
1. Identify the given equation:
[tex]\[
\frac{x}{x-4} - \frac{8}{4-x} = 2
\][/tex]
2. Simplify the second term:
Notice that [tex]\(4 - x\)[/tex] can be rewritten as [tex]\(-(x - 4)\)[/tex], so:
[tex]\[
\frac{8}{4-x} = \frac{8}{-(x-4)} = -\frac{8}{x-4}
\][/tex]
Substituting this into the original equation, we get:
[tex]\[
\frac{x}{x-4} - \left(-\frac{8}{x-4}\right) = 2
\][/tex]
Simplifying further:
[tex]\[
\frac{x}{x-4} + \frac{8}{x-4} = 2
\][/tex]
3. Combine the fractions:
Both fractions have the same denominator. Therefore, we can combine them:
[tex]\[
\frac{x + 8}{x-4} = 2
\][/tex]
4. Clear the fraction by multiplying both sides by the denominator:
Multiply both sides by [tex]\(x-4\)[/tex]:
[tex]\[
x + 8 = 2(x - 4)
\][/tex]
5. Simplify the resulting equation:
Distribute the 2 on the right-hand side:
[tex]\[
x + 8 = 2x - 8
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
8 + 8 = 2x - x
\][/tex]
[tex]\[
16 = x
\][/tex]
Therefore, the solution to the equation [tex]\(\frac{x}{x-4} - \frac{8}{4-x} = 2\)[/tex] is:
[tex]\[
x = 16
\][/tex]
