To solve the equation [tex]\(\frac{2x}{x-1} + \frac{3x}{x+2} = 5\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Find a common denominator for the fractions:
The common denominator for [tex]\(x - 1\)[/tex] and [tex]\(x + 2\)[/tex] is [tex]\((x - 1)(x + 2)\)[/tex].
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{2x}{x-1} = \frac{2x(x+2)}{(x-1)(x+2)}
\][/tex]
[tex]\[
\frac{3x}{x+2} = \frac{3x(x-1)}{(x-1)(x+2)}
\][/tex]
3. Combine the fractions:
[tex]\[
\frac{2x(x+2) + 3x(x-1)}{(x-1)(x+2)} = 5
\][/tex]
4. Simplify the numerator:
[tex]\[
2x(x+2) = 2x^2 + 4x
\][/tex]
[tex]\[
3x(x-1) = 3x^2 - 3x
\][/tex]
[tex]\[
2x^2 + 4x + 3x^2 - 3x = 5x^2 + x
\][/tex]
So, the equation becomes:
[tex]\[
\frac{5x^2 + x}{(x-1)(x+2)} = 5
\][/tex]
5. Eliminate the denominator by multiplying both sides by [tex]\((x-1)(x+2)\)[/tex]:
[tex]\[
5x^2 + x = 5(x-1)(x+2)
\][/tex]
6. Expand the right side:
[tex]\[
5(x-1)(x+2) = 5(x^2 + 2x - x - 2) = 5(x^2 + x - 2) = 5x^2 + 5x - 10
\][/tex]
7. Set the equation to zero:
[tex]\[
5x^2 + x = 5x^2 + 5x - 10
\][/tex]
[tex]\[
5x^2 + x - 5x^2 - 5x + 10 = 0
\][/tex]
[tex]\[
-4x + 10 = 0
\][/tex]
8. Solve for [tex]\(x\)[/tex]:
[tex]\[
-4x + 10 = 0
\][/tex]
[tex]\[
-4x = -10
\][/tex]
[tex]\[
x = \frac{10}{4}
\][/tex]
[tex]\[
x = \frac{5}{2}
\][/tex]
So, the solution to the equation [tex]\(\frac{2x}{x-1} + \frac{3x}{x+2} = 5\)[/tex] is [tex]\(x = \frac{5}{2}\)[/tex].
