Question

Solve for x 2/x+2 + 1/5 = 6/x+5 x = 0 x = –13 x = 13 x = 0 and x = 13

Answer 1

$\dfrac{2}{x+2}+\dfrac{1}{5}=\dfrac{6}{x+5}\qquad\text{multiply both sides by 5}\\\\\dfrac{10}{x+2}+1=\dfrac{30}{x+5}\\\\\dfrac{10}{x+2}+\dfrac{x+2}{x+2}=\dfrac{30}{x+5}\\\\\dfrac{10+x+2}{x+2}=\dfrac{30}{x+5}\\\\\dfrac{x+12}{x+2}=\dfrac{30}{x+5}\qquad\text{cross multiply}\\\\(x+12)(x+5)=30(x+2)\qquad\text{use distributive property}\\\\(x)(x)+(x)(5)+(12)(x)+(12)(5)=(30)(x)+(30)(2)\\\\x^2+5x+12x+60=30x+60\qquad\text{subtract 60 from both sides}\\\\x^2+17x=30x\qquad\text{subtract 30x from both sides}$

$x^2-13x=0\\\\x(x-13)=0\iff x=0\ \vee\ x-13=0\qquad\text{add 13 to both sides}\\\\\boxed{x=0\ \vee\ x=13}$