Question

adding and subtracting rational expressions with three terms
simplify: \\(\frac{x}{x - 3} - \frac{3}{x + 4} + \frac{7}{x^2 + x - 12}\\)
rewrite the expression using the lcd.
\\(\frac{x(x + 4) + 3(x - 3) - 7}{(x - 3)(x + 4)}\\)
\\(\frac{x(x + 4) - 3(x - 3) + 7}{(x - 3)(x + 4)}\\)
\\(\frac{x(x + 4) + 3(x - 3) + 7}{(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 \(x^2 + x - 12=(x + 4)(x - 3)\).

Step2: Find the LCD

The denominators are \(x - 3\), \(x + 4\), and \((x + 4)(x - 3)\). The least common denominator (LCD) is \((x - 3)(x + 4)\).

Step3: Rewrite Each Term

• For \(\frac{x}{x - 3}\), multiply numerator and denominator by \(x + 4\): \(\frac{x(x + 4)}{(x - 3)(x + 4)}\)
• For \(\frac{3}{x + 4}\), the subtraction means we multiply numerator and denominator by \(x - 3\) and keep the negative: \(\frac{-3(x - 3)}{(x - 3)(x + 4)}\) (which is equivalent to \(\frac{3(-(x - 3))}{(x - 3)(x + 4)}\) or \(\frac{-3(x - 3)}{(x - 3)(x + 4)}\), but when combining, it's \(-3(x - 3)\) in the numerator)
• For \(\frac{7}{x^2 + x - 12}=\frac{7}{(x - 3)(x + 4)}\), the denominator is already the LCD, so the numerator remains \(7\).

Now, combine the numerators over the LCD: \(\frac{x(x + 4)-3(x - 3)+7}{(x - 3)(x + 4)}\)

Answer:

\(\frac{x(x + 4)-3(x - 3)+7}{(x - 3)(x + 4)}\) (the second option in the given choices)