Question

Solve for x 4/x + 4/x^2-9= 3/x-3

x = –4 and x = –9
x = 4 and x = –9
x = –4 and x = 9
x = 4 and x = 9

Answer 1

`(4)/(x)+(4)/(x^2-9)=(3)/(x-3)\qquad\text{subtract}\ (4)/(x^2-9)\ \text{from both sides}\\\\(4)/(x)=(3)/(x-3)-(4)/(x^2-9)\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\\\(4)/(x)=(3)/(x-3)-(4)/(x^2-3^2)\\\\(4)/(x)=(3)/(x-3)-(4)/((x-3)(x+3))\\\\(4)/(x)=(3(x+3))/((x-3)(x+3))-(4)/((x-3)(x+3))\qquad\text{use distributive property}\\\\(4)/(x)=(3x+9-4)/((x-3)(x+3))\\\\(4)/(x)=(3x+5)/(x^2-9)\qquad\text{cross multiply}`

`4(x^2-9)=x(3x+5)\qquad\text{use distributive property}\\\\4x^2-36=3x^2+5x\qquad\text{subtract}\ 3x^2\ \text{and}\ 5x\ \text{from both sides}\\\\x^2-5x-36=0\\\\x^2-9x+4x-36=0\\\\x(x-9)+4(x-9)=0\\\\(x-9)(x+4)=0\iff x-9=0\ \vee\ x+4=0\\\\\boxed{x=9\ \vee\ x=-4}`