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 + 2} - \frac{1}{x - 2} = \frac{3x}{x^2 - 4}\\)

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: Analyze denominators

The denominators are \(x + 2\), \(x - 2\), and \(x^2 - 4\). Factor \(x^2 - 4\) as \((x + 2)(x - 2)\) (using the difference of squares formula \(a^2 - b^2=(a + b)(a - b)\) where \(a = x\) and \(b = 2\)).

Step2: Find values that make denominators zero

• For \(x + 2 = 0\), solve for \(x\): \(x=-2\).
• For \(x - 2 = 0\), solve for \(x\): \(x = 2\).
• For \((x + 2)(x - 2)=0\), by the zero - product property, either \(x + 2 = 0\) (which gives \(x=-2\)) or \(x - 2 = 0\) (which gives \(x = 2\)).

Answer:

-2, 2