Answer
The solution set is $\left \{-\frac{6}{5},4 \right \}$.
Work Step by Step
The given expression is
$\frac{6}{x}+\frac{6}{x+2}=\frac{5}{2}$
The lowest common multiple of denominators is
$2x(x+2)$
Multiply all fractions by lowest common multiple.
$[2x(x+2)]\frac{6}{x}+[2x(x+2)]\frac{6}{x+2}=[2x(x+2)]\frac{5}{2}$
Simplify.
$12(x+2)+12x=5x(x+2)$
Clear the parentheses.
$12x+24+12x=5x^2+10x$
$24+24x=5x^2+10x$
$0=5x^2+10x-24x-24$
$0=5x^2+14x-24$
Factor.
$0=5x^2+6x-20x-24$
$0=x(5x+6)-4(5x+6)$
$0=(5x+6)(x-4)$
Equate both factors equal to zero.
$5x+6=0$ or $x-4=0$
$5x=-6$ or $x=4$
$x=-\frac{6}{5}$ or $x=4$.
