To simplify the expression [tex]\(\frac{4}{2-x} - \frac{3}{x}\)[/tex], follow these steps:
1. Identify the least common denominator (LCD):
The denominators are [tex]\(2 - x\)[/tex] and [tex]\(x\)[/tex]. The LCD is the product of these two denominators, which is [tex]\(x(2 - x)\)[/tex].
2. Rewrite each fraction with the LCD as the new denominator:
For the first fraction [tex]\(\frac{4}{2-x}\)[/tex], multiply the numerator and denominator by [tex]\(x\)[/tex]:
[tex]\[
\frac{4}{2-x} \cdot \frac{x}{x} = \frac{4x}{x(2-x)}
\][/tex]
For the second fraction [tex]\(\frac{3}{x}\)[/tex], multiply the numerator and denominator by [tex]\(2 - x\)[/tex]:
[tex]\[
\frac{3}{x} \cdot \frac{2-x}{2-x} = \frac{3(2-x)}{x(2-x)}
\][/tex]
3. Combine the fractions over the common denominator:
[tex]\[
\frac{4x}{x(2-x)} - \frac{3(2-x)}{x(2-x)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{4x - 3(2 - x)}{x(2 - x)}
\][/tex]
4. Simplify the numerator:
Expand the numerator:
[tex]\[
4x - 3(2 - x) = 4x - 6 + 3x = 4x + 3x - 6 = 7x - 6
\][/tex]
5. Write the final simplified expression:
[tex]\[
\frac{7x - 6}{x(2 - x)}
\][/tex]
Thus, the simplified form of the given expression [tex]\(\frac{4}{2-x} - \frac{3}{x}\)[/tex] is:
[tex]\[
\frac{6 - 7x}{x(x - 2)}
\][/tex]
Note that the change of sign in the final simplified form of the expression can be achieved by swapping [tex]\(2-x\)[/tex] to [tex]\(-(x-2)\)[/tex]:
[tex]\[
\frac{-(7x - 6)}{-x(x - 2)} = \frac{6 - 7x}{x(x - 2)}
\][/tex]
So, the final answer is indeed:
[tex]\[
\frac{6 - 7x}{x(x - 2)}
\][/tex]
