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 + 12} - \frac{3}{x - 12} = \frac{5x}{x^2 - 144}\\)

a. write the value or values of the variable that make a denominator zero.
x = -12,12 (use a comma to separate answers as needed.)

b. what is the solution of the equation? select the correct choice below and, if necessary, fill in the answer box to complete your choice

a. the solution set is {}. (use a comma to separate answers as needed.)

b. the solution set is {x | x is a real number}

c. the solution set is \\(\varnothing\\)

Explanation:

Step1: Factor the right denominator

Notice that $x^2 - 144 = (x+12)(x-12)$

Step2: Find variable restrictions

Set each denominator equal to 0:
$x+12=0 \implies x=-12$
$x-12=0 \implies x=12$

Step3: Eliminate denominators

Multiply all terms by $(x+12)(x-12)$:
$4(x-12) - 3(x+12) = 5x$

Step4: Expand and simplify left side

$4x - 48 - 3x - 36 = 5x$
$x - 84 = 5x$

Step5: Solve for x

$x - 5x = 84$
$-4x = 84$
$x = \frac{84}{-4} = -21$

Step6: Check against restrictions

$-21$ is not equal to $-12$ or $12$, so it is valid.

Answer:

a. $x=-12,12$
b. A. The solution set is $\{-21\}$