Simplify the following complex rational expression. $$ \frac{-\frac{6}{x+4}+\frac{2}{x-5}}{-\frac{3}{x+4}} $$

Question

UpStudy AI · Accepted Answer

To simplify the complex rational expression:

$$
\frac{-\frac{6}{x+4} + \frac{2}{x-5}}{-\frac{3}{x+4}}
$$

**Step 1: Simplify the Numerator**

Combine the terms in the numerator:

$$
-\frac{6}{x+4} + \frac{2}{x-5} = \frac{-6(x-5) + 2(x+4)}{(x+4)(x-5)}
$$

Expand and combine like terms:

$$
\frac{-6x + 30 + 2x + 8}{(x+4)(x-5)} = \frac{-4x + 38}{(x+4)(x-5)}
$$

**Step 2: Form the Complete Fraction**

Now, the expression becomes:

$$
\frac{\frac{-4x + 38}{(x+4)(x-5)}}{\frac{-3}{x+4}}
$$

**Step 3: Divide by a Fraction (Multiply by its Reciprocal)**

$$
\frac{-4x + 38}{(x+4)(x-5)} \div \frac{-3}{x+4} = \frac{-4x + 38}{(x+4)(x-5)} \times \frac{x+4}{-3}
$$

**Step 4: Simplify by Canceling Common Factors**

Cancel out the $$(x+4)$$ terms:

$$
\frac{-4x + 38}{x-5} \times \frac{1}{-3} = \frac{(-4x + 38)(-1)}{3(x-5)} = \frac{4x - 38}{3(x-5)}
$$

**Final Simplified Expression:**

$$
\frac{4x - 38}{3(x - 5)}
$$

**Alternatively, factoring the numerator:**

$$
\frac{2(2x - 19)}{3(x - 5)}
$$

Either form is correct, but the simplified form without factoring is often preferred.

**Answer:**

After simplifying, the expression is (4 x – 38) divided by [3 (x – 5)]. Thus,

(4x−38)/(3(x−5))
The simplified form of the expression is $$\frac{4x - 38}{3(x - 5)}$$.

Simplify the following complex rational expression. \[ \frac{-\frac{6}{x+4}+\frac{2}{x-5}}{-\frac{3}{x+4}} \]