Answer
No solution or $\varnothing $.
Work Step by Step
First we clear the fractions by multiplying each term by the Least Common Denominator.
Factor $x^2-9$.
$=x^2-3^2$
Use the algebraic identity $a^2-b^2=(a+b)(a-b)$.
$=(x+3)(x-3)$
Substitute factor into the given equation.
$\Rightarrow \frac{4}{x-3}-\frac{6}{x+3}=\frac{24}{(x+3)(x-3)}$
Multiply the equation by $LCD =(x+3)(x-3)$.
$\Rightarrow (x+3)(x-3)\left (\frac{4}{x-3}-\frac{6}{x+3}\right )=(x+3)(x-3)\left (\frac{24}{(x+3)(x-3)}\right )$
Use the distributive property.
$\Rightarrow (x+3)(x-3)\cdot \frac{4}{x-3}-(x+3)(x-3)\cdot\frac{6}{x+3}=(x+3)(x-3)\cdot \frac{24}{(x+3)(x-3)}$
Cancel common terms.
$\Rightarrow 4(x+3)-6(x-3)=24$
Use distributive property.
$\Rightarrow 4x+12-6x+18=24$
Simplify.
$\Rightarrow -2x+30=24$
Subtract $30$ from both sides.
$\Rightarrow -2x+30-30=24-30$
Simplify.
$\Rightarrow -2x=-6$
Divide both sides by $-2$.
$\Rightarrow \frac{-2x}{-2}=\frac{-6}{-2}$
Simplify.
$\Rightarrow x=3$
Verify solution.
Plug $x=3$ into the given equation.
$\Rightarrow \frac{4}{3-3}-\frac{6}{3+3}=\frac{24}{3^2-9}$
The first term on the left side and the term on the right side are undefined.
Hence, the equation has no solution.
