To express [tex]\(\frac{3}{x} + \frac{5}{x-2}\)[/tex] as a single fraction in its simplest form, we need to find a common denominator. Here are the steps to do that:
1. Identify the denominators:
The denominators are [tex]\(x\)[/tex] and [tex]\(x-2\)[/tex].
2. Find the common denominator:
The common denominator will be the product of the two denominators, i.e., [tex]\(x(x-2)\)[/tex].
3. Rewrite each fraction with the common denominator:
We need to adjust each fraction so that they have the same denominator [tex]\(x(x-2)\)[/tex].
- For [tex]\(\frac{3}{x}\)[/tex], multiply the numerator and the denominator by [tex]\((x-2)\)[/tex]:
[tex]\[
\frac{3}{x} = \frac{3 \cdot (x-2)}{x \cdot (x-2)} = \frac{3(x-2)}{x(x-2)}
\][/tex]
- For [tex]\(\frac{5}{x-2}\)[/tex], multiply the numerator and the denominator by [tex]\(x\)[/tex]:
[tex]\[
\frac{5}{x-2} = \frac{5 \cdot x}{(x-2) \cdot x} = \frac{5x}{x(x-2)}
\][/tex]
4. Combine the fractions:
Now that both fractions have the common denominator [tex]\(x(x-2)\)[/tex], we can combine them into a single fraction:
[tex]\[
\frac{3(x-2)}{x(x-2)} + \frac{5x}{x(x-2)} = \frac{3(x-2) + 5x}{x(x-2)}
\][/tex]
5. Simplify the numerator:
Expand and combine the terms in the numerator:
[tex]\[
3(x-2) + 5x = 3x - 6 + 5x = 8x - 6
\][/tex]
6. Write the final simplified fraction:
So the combined fraction in its simplest form is:
[tex]\[
\frac{8x - 6}{x(x-2)} = \frac{2(4x - 3)}{x(x-2)}
\][/tex]
Therefore, the fraction [tex]\(\frac{3}{x} + \frac{5}{x-2}\)[/tex] expressed as a single fraction in its simplest form is:
[tex]\[
\frac{8x - 6}{x(x-2)} \quad \text{or equivalently} \quad \frac{2(4x - 3)}{x(x-2)}.
\][/tex]
