Certainly! Let's simplify the expression [tex]\(\frac{3}{2x} + \frac{5}{x-2}\)[/tex] step by step.
1. Combine the fractions:
To combine the fractions [tex]\(\frac{3}{2x}\)[/tex] and [tex]\(\frac{5}{x-2}\)[/tex], first find a common denominator. The least common denominator (LCD) of [tex]\(2x\)[/tex] and [tex]\(x-2\)[/tex] is [tex]\(2x(x-2)\)[/tex].
2. Rewrite each fraction with the common denominator:
- For the first fraction, [tex]\(\frac{3}{2x}\)[/tex]:
[tex]\[
\frac{3}{2x} = \frac{3(x-2)}{2x(x-2)}
\][/tex]
- For the second fraction, [tex]\(\frac{5}{x-2}\)[/tex]:
[tex]\[
\frac{5}{x-2} = \frac{5 \cdot 2x}{2x(x-2)} = \frac{10x}{2x(x-2)}
\][/tex]
3. Add the fractions:
Now that both fractions have the same denominator, we can add them:
[tex]\[
\frac{3(x-2)}{2x(x-2)} + \frac{10x}{2x(x-2)} = \frac{3(x-2) + 10x}{2x(x-2)}
\][/tex]
4. Simplify the numerator:
Distribute the 3 in the first term of the numerator:
[tex]\[
3(x-2) = 3x - 6
\][/tex]
Combine the terms in the numerator:
[tex]\[
3(x-2) + 10x = 3x - 6 + 10x = 13x - 6
\][/tex]
5. Write the final simplified expression:
Putting it all together, we get:
[tex]\[
\frac{13x - 6}{2x(x-2)}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{13x - 6}{2x(x-2)}
\][/tex]
