Question

for the following equation, a. write the value or values of the variable that make a denominator zero. these are the restrictions on the variable. b. keeping the restrictions in mind, solve the equation.
\\(\frac{2}{x + 9} - \frac{3}{x - 9} = \frac{2x}{x^2 - 81}\\)

a. write the value or values of the variable that make a denominator zero.
\\(x = \square\\) (use a comma to separate answers as needed.)

Explanation:

Step1: Identify denominators

The denominators are \(x + 9\), \(x - 9\), and \(x^2 - 81\). Note that \(x^2 - 81=(x + 9)(x - 9)\) by difference of squares.

Step2: Set each denominator to zero

• For \(x + 9 = 0\), solve for \(x\): \(x=-9\).
• For \(x - 9 = 0\), solve for \(x\): \(x = 9\).
• For \(x^2 - 81 = 0\), we already know from the above that \(x=\pm9\) (since \(x^2 - 81=(x + 9)(x - 9)\)).

Answer:

-9, 9