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: Factor the quadratic denominator

$x^2 - 256 = (x+16)(x-16)$

Step2: Find zeros of denominators

Set $x+16=0 \implies x=-16$; set $x-16=0 \implies x=16$

Step3: Verify restricted values

Both $x=-16$ and $x=16$ make denominators zero.

Step4: Clear denominators to solve equation

Multiply all terms by $(x+16)(x-16)$:
$4(x-16) - 3(x+16) = 5x$

Step5: Expand and simplify left side

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

Step6: Isolate the variable $x$

$x - 5x = 112$
$-4x = 112$
$x = -28$

Step7: Check against restrictions

$x=-28$ is not $-16$ or $16$, so it is valid.

Answer:

a. $x = -16, 16$
b. $x = -28$