Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
adding and subtracting rational expressions
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.
options:

1. (\frac{x(x + 4) + 3(x - 3) + 7}{(x - 3)(x + 4)})
2. (\frac{x(x + 4) + 3(x - 3) - 7}{(x - 3)(x + 4)})
3. (\frac{x(x + 4) - 3(x - 3) + 7}{(x - 3)(x + 4)})

Explanation:

Step1: Factor the denominator of the third term

First, 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 fraction with the LCD

• For \(\frac{x}{x - 3}\), multiply numerator and denominator by \(x + 4\): \(\frac{x(x + 4)}{(x - 3)(x + 4)}\)
• For \(\frac{3}{x + 4}\), multiply numerator and denominator by \(x - 3\): \(\frac{3(x - 3)}{(x - 3)(x + 4)}\)
• The third term \(\frac{7}{(x + 4)(x - 3)}\) already has the LCD, so it remains \(\frac{7}{(x - 3)(x + 4)}\)

Step4: Combine the fractions

Now we have:
\[
\frac{x(x + 4)}{(x - 3)(x + 4)}-\frac{3(x - 3)}{(x - 3)(x + 4)}+\frac{7}{(x - 3)(x + 4)}
\]
Combine the numerators over the common denominator:
\[
\frac{x(x + 4)-3(x - 3)+7}{(x - 3)(x + 4)}
\]

Answer:

\(\frac{x(x + 4)-3(x - 3)+7}{(x - 3)(x + 4)}\) (which matches the first option's numerator structure when expanded, but the key is the rewritten form with LCD)