Simplify. $$ \frac{6}{x-2}-\frac{x+7}{x+2} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{6}{x-2} - \frac{x+7}{x+2}
$$

**Step 1: Find a Common Denominator**

The common denominator for the two fractions is $$(x - 2)(x + 2)$$.

**Step 2: Rewrite Each Fraction with the Common Denominator**

$$
\frac{6}{x-2} = \frac{6(x + 2)}{(x - 2)(x + 2)}
$$
$$
\frac{x+7}{x+2} = \frac{(x + 7)(x - 2)}{(x - 2)(x + 2)}
$$

**Step 3: Combine the Fractions**

$$
\frac{6(x + 2) - (x + 7)(x - 2)}{(x - 2)(x + 2)}
$$

**Step 4: Expand and Simplify the Numerator**

First, expand both terms in the numerator:

$$
6(x + 2) = 6x + 12
$$
$$
(x + 7)(x - 2) = x^2 + 7x - 2x - 14 = x^2 + 5x - 14
$$

Now, subtract the second expanded term from the first:

$$
6x + 12 - (x^2 + 5x - 14) = 6x + 12 - x^2 - 5x + 14 = -x^2 + x + 26
$$

**Step 5: Write the Final Simplified Expression**

$$
\frac{-x^2 + x + 26}{(x - 2)(x + 2)}
$$

This can also be written as:

$$
\frac{-x^2 + x + 26}{x^2 - 4}
$$

**Final Answer:**

$$
\boxed{\dfrac{-\,x^{2} + x + 26}{\,x^{2} - 4\,}}
$$
$$ \frac{-x^2 + x + 26}{x^2 - 4} $$

Simplify.