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{4}{x + 16} - \frac{3}{x - 16} = \frac{5x}{x^2 - 256}\\)

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: Find zero of first denominator

Set $x+16=0$, solve for $x$:
$x=-16$

Step2: Find zero of second denominator

Set $x-16=0$, solve for $x$:
$x=16$

Step3: Factor third denominator

$x^2-256=(x+16)(x-16)$, so its zeros are the same as above: $x=-16, 16$

(for solving the equation, part b):

Step1: Identify common denominator

Common denominator is $(x+16)(x-16)=x^2-256$

Step2: Multiply all terms by denominator

$$4(x-16) - 3(x+16) = 5x$$

Step3: Expand and simplify left side

$$4x-64-3x-48=5x$$
$$x-112=5x$$

Step4: Isolate $x$ terms

$$x-5x=112$$
$$-4x=112$$

Step5: Solve for $x$

$$x=\frac{112}{-4}=-28$$

Step6: Check against restrictions

$-28
eq -16, 16$, so it is valid.

Answer:

$x=-16, 16$

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