Let's simplify the expression [tex]\(\frac{x}{2(x+y)} - \frac{2}{3(x+y)}\)[/tex].
1. Identify the common denominator:
The denominators of the two fractions are [tex]\(2(x + y)\)[/tex] and [tex]\(3(x + y)\)[/tex], respectively. To combine these fractions, we need a common denominator. In this case, the common denominator is the product of [tex]\(2\)[/tex] and [tex]\(3\)[/tex], which is [tex]\(6\)[/tex], multiplied by [tex]\((x + y)\)[/tex]. Thus, the common denominator is [tex]\(6(x + y)\)[/tex].
2. Rewrite each fraction with the common denominator:
- For the first fraction [tex]\(\frac{x}{2(x + y)}\)[/tex], multiply both the numerator and the denominator by [tex]\(3\)[/tex]:
[tex]\[
\frac{x}{2(x + y)} \cdot \frac{3}{3} = \frac{3x}{6(x + y)}
\][/tex]
- For the second fraction [tex]\(\frac{2}{3(x + y)}\)[/tex], multiply both the numerator and the denominator by [tex]\(2\)[/tex]:
[tex]\[
\frac{2}{3(x + y)} \cdot \frac{2}{2} = \frac{4}{6(x + y)}
\][/tex]
3. Combine the fractions:
Now that both fractions have the same denominator, we can combine them:
[tex]\[
\frac{3x}{6(x + y)} - \frac{4}{6(x + y)} = \frac{3x - 4}{6(x + y)}
\][/tex]
4. Further simplification (if possible):
We can leave the expression as it is or transform it into another form. Notice that the numerator [tex]\(3x - 4\)[/tex] cannot be further simplified with the denominator [tex]\(6(x + y)\)[/tex].
Alternatively, let's explore another method of expressing it:
We can rewrite [tex]\(\frac{3x - 4}{6(x + y)}\)[/tex] as a sum of two separate fractions:
[tex]\[
\frac{3x}{6(x + y)} - \frac{4}{6(x + y)} = \frac{x}{2(x + y)} - \frac{2}{3(x + y)}
\][/tex]
Observing the numerator [tex]\(3x - 4\)[/tex], we can also convert it into:
[tex]\[
\frac{3x - 4}{6(x + y)} = \frac{1}{6(x + y)}(3x - 4)
\][/tex]
To further simplify, we can distribute the terms to split the fraction:
[tex]\[
\frac{3x}{6(x + y)} - \frac{4}{6(x + y)} = \left(\frac{x}{2}\right) \cdot \left( \frac{1}{(x + y)} \right) - \left(\frac{2}{3}\right) \cdot \left( \frac{1}{(x + y)} \right)
\][/tex]
Combine into:
[tex]\[
\frac{\frac{x}{2} - \frac{2}{3}}{(x + y)}
\][/tex]
Simplifying the fractions inside:
[tex]\[
= \frac{x/2 - 2/3}{x + y}
\][/tex]
Thus, the expression simplifies to:
[tex]\[
\boxed{\frac{x/2 - 2/3}{x + y}}
\][/tex]
