Question

solve.
$\frac{16}{x^{2}-4}-\frac{4}{x-2}=\frac{7}{x+2}$
select the correct choice and, if necessary, fill in the answer box in your choice below.
a. the solution set is {}.
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
b. the solution is the empty set.

Explanation:

Step1: Factor denominator

Note that $x^2-4=(x-2)(x+2)$.

Step2: Find common denominator

Multiply all terms by $(x-2)(x+2)$ to eliminate fractions:
$$16 - 4(x+2) = 7(x-2)$$

Step3: Expand parentheses

$$16 - 4x - 8 = 7x - 14$$

Step4: Simplify left side

$$8 - 4x = 7x - 14$$

Step5: Isolate x terms

$$8 + 14 = 7x + 4x$$
$$22 = 11x$$

Step6: Solve for x

$$x = \frac{22}{11} = 2$$

Step7: Check for extraneous solutions

Substitute $x=2$ into original equation: denominators $x-2=0$ and $x^2-4=0$, which makes the equation undefined. So $x=2$ is not a valid solution.

Answer:

B. The solution is the empty set.