Question

Solve this rational equation ​

Answer 1

x = -1

Explanation:

`\text{Domain:}\\\\x-4\\eq0\ \wedge\ x-2\\eq0\ \wedge\ x^2-6x+8\\eq0\\\\x\\eq4\ \wedge\ x\\eq2\ \wedge\ x^2-4x-2x+8\\eq0\\\\x\\eq4\ \wedge\ x\\eq2\ \wedge\ x(x-4)-2(x-4)\\eq0\\\\x\\eq4\ \wedge\ x\\eq2\ \wedge\ (x-4)(x-2)\\eq0\\\\x\\eq4\ \wedge\ x\\eq2\ \wedge\ x-4\\eq0\ \wedge\ x-2\\eq0\\\\\boxed{D:x\\eq4\ \wedge\ x\\eq2}`

`(1)/(x-4)+(x)/(x-2)=(2)/(x^2-6x+8)\\\\(1(x-2))/((x-4)(x-2))+(x(x-4))/((x-4)(x-2))=(2)/((x-4)(x-2))\\\\(x-2+x(x-4))/((x-4)(x-2))=(2)/((x-4)(x-2))\iff x-2+x(x-4)=2\\\\x-2+(x)(x)+(x)(-4)=2\\\\x-2+x^2-4x=2\qquad\text{subtract 2 from both sides}\\\\x^2-3x-4=0\\\\x^2-4x+x-4=0\\\\x(x-4)+1(x-4)=0\\\\(x-4)(x+1)=0\iff x-4=0\ \wedge\ x+1=0\\\\x-4=0\qquad\text{add 4 to both sides}\\x=4\\otin D\\\\x+1=0\qquad\text{subtract 1 from both sides}\\x=-1\in D`