Question

solve.
\\(\frac{3}{x + 5} - \frac{2}{x - 5} = \frac{6x}{x^2 - 25}\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. \\(x = -5\\)
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
b. the solution is all real numbers.
c. there is no solution.

Explanation:

Step1: Factor the denominator

Notice that \(x^2 - 25\) is a difference of squares, so \(x^2 - 25=(x + 5)(x - 5)\). The equation becomes \(\frac{3}{x + 5}-\frac{2}{x - 5}=\frac{6x}{(x + 5)(x - 5)}\).

Step2: Multiply through by the LCD

The least common denominator (LCD) of the fractions is \((x + 5)(x - 5)\). Multiply each term by \((x + 5)(x - 5)\) to eliminate the denominators:

$$
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Step3: Solve for \(x\)

Subtract \(x\) from both sides:

$$ -25 = 5x $$

Divide both sides by 5:

$$ x=- 5 $$

Step4: Check for extraneous solutions

Now, check if \(x = - 5\) makes the original denominators zero. For the denominators \(x + 5\) and \(x^2-25\), when \(x=-5\), \(x + 5=-5 + 5 = 0\) and \(x^2-25=(-5)^2-25=25 - 25 = 0\). So \(x=-5\) is an extraneous solution, which means there is no solution to the equation.

Answer:

C. There is no solution