Question

write as a single fraction in its simplest form.
\\(\frac{3}{x - 4} - \frac{4}{x + 3}\\)

Explanation:

Step1: Find a common denominator

The denominators are \(x - 4\) and \(x + 3\), so the common denominator is \((x - 4)(x + 3)\).
Rewrite each fraction with the common denominator:
\(\frac{3}{x - 4}=\frac{3(x + 3)}{(x - 4)(x + 3)}\)
\(\frac{4}{x + 3}=\frac{4(x - 4)}{(x - 4)(x + 3)}\)

Step2: Subtract the fractions

Now subtract the two fractions:
\[

$$\begin{align*} \frac{3(x + 3)}{(x - 4)(x + 3)}-\frac{4(x - 4)}{(x - 4)(x + 3)}&=\frac{3(x + 3)-4(x - 4)}{(x - 4)(x + 3)}\\ \end{align*}$$

\]

Step3: Expand and simplify the numerator

Expand the numerator:
\(3(x + 3)-4(x - 4)=3x + 9 - 4x + 16\)
Combine like terms:
\(3x + 9 - 4x + 16=-x + 25\)

Step4: Write the final fraction

So the expression becomes:
\(\frac{-x + 25}{(x - 4)(x + 3)}\) or we can factor out a negative sign from the numerator: \(\frac{-(x - 25)}{(x - 4)(x + 3)}=\frac{25 - x}{(x - 4)(x + 3)}\)

Answer:

\(\frac{25 - x}{(x - 4)(x + 3)}\) (or \(\frac{-x + 25}{(x - 4)(x + 3)}\))