Question

adding and subtracting rational expressions with three terms
simplify: \\(\frac{x}{x - 3} - \frac{3}{x + 4} + \frac{7}{x^2 + x - 12}\\)
options: \\(\frac{x^2 + x + 4}{(x - 3)(x + 4)}\\), \\(\frac{x^2 + x + 16}{(x - 3)(x + 4)}\\), \\(\frac{x^2 - 3x + 8}{(x - 3)(x + 4)}\\)

Explanation:

Step1: Factor the Denominator

Factor \(x^2 + x - 12\). We need two numbers that multiply to \(-12\) and add to \(1\). Those numbers are \(4\) and \(-3\). So, \(x^2 + x - 12=(x + 4)(x - 3)\).
The expression becomes \(\frac{x}{x - 3}-\frac{3}{x + 4}+\frac{7}{(x - 3)(x + 4)}\).

Step2: Find a Common Denominator

The common denominator for the three fractions is \((x - 3)(x + 4)\).
Rewrite each fraction with the common denominator:

• For \(\frac{x}{x - 3}\), multiply numerator and denominator by \((x + 4)\): \(\frac{x(x + 4)}{(x - 3)(x + 4)}=\frac{x^2 + 4x}{(x - 3)(x + 4)}\)
• For \(\frac{3}{x + 4}\), multiply numerator and denominator by \((x - 3)\): \(\frac{3(x - 3)}{(x - 3)(x + 4)}=\frac{3x - 9}{(x - 3)(x + 4)}\)
• The third fraction \(\frac{7}{(x - 3)(x + 4)}\) already has the common denominator.

Step3: Combine the Fractions

Now, combine the numerators over the common denominator:

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Answer:

\(\frac{x^2 + x + 16}{(x - 3)(x + 4)}\) (the middle option)