Question

1/x-3+2/x-2=8/x,x≠4,7​

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:x\:=\:\dfrac{12}{5}\:}}}$

Step-by-step-explanation:

We have given a equation in variable x.

We have to find the value of x.

Now,

$\displaystyle{\sf\:\dfrac{1}{x\:-\:3}\:+\:\dfrac{2}{x\:-\:2}\:=\:\dfrac{8}{x}}$

$\displaystyle{\implies\sf\:\dfrac{x\:-\:2\:+\:2\:(\:x\:-\:3\:)}{(\:x\:-\:3\:)\:(\:x\:-\:2\:)}\:=\:\dfrac{8}{x}}$

$\displaystyle{\implies\sf\:\dfrac{x\:-\:2\:+\:2x\:-\:6}{x^2\:-\:2x\:-\:3x\:+\:6}\:=\:\dfrac{8}{x}}$

$\displaystyle{\implies\sf\:\dfrac{x\:+\:2x\:-\:2\:-\:6}{x^2\:-\:5x\:+\:6}\:=\:\dfrac{8}{x}}$

$\displaystyle{\implies\sf\:\dfrac{3x\:-\:8}{x^2\:-\:5x\:+\:6}\:=\:\dfrac{8}{x}}$

$\displaystyle{\implies\sf\:(\:3x\:-\:8\:)\:\times\:x\:=\:8\:\times\:(\:x^2\:-\:5x\:+\:6\:)}$

$\displaystyle{\implies\sf\:3x^2\:-\:8x\:=\:8x^2\:-\:40x\:+\:48}$

$\displaystyle{\implies\sf\:8x^2\:-\:40x\:+\:48\:-\:3x^2\:+\:8x\:=\:0}$

$\displaystyle{\implies\sf\:8x^2\:-\:3x^2\:-\:40x\:+\:8x\:+\:48\:=\:0}$

$\displaystyle{\implies\sf\:5x^2\:-\:32x\:+\:48\:=\:0}$

$\displaystyle{\implies\sf\:5x^2\:-\:20x\:-\:12x\:+\:48\:=\:0}$

$\displaystyle{\implies\sf\:5x\:(\:x\:-\:4\:)\:-\:12\:(\:x\:-\:4\:)\:=\:0}$

$\displaystyle{\implies\sf\:(\:x\:-\:4\:)\:(\:5x\:-\:12\:)\:=\:0}$

$\displaystyle{\implies\sf\:(\:x\:-\:4\:)\:=\:0\:\quad\:OR\:\quad\:(\:5x\:-\:12\:)\:=\:0}$

$\displaystyle{\implies\sf\:x\:-\:4\:=\:0\:\quad\:OR\:\quad\:5x\:-\:12\:=\:0}$

$\displaystyle{\implies\sf\:x\:=\:4\:\quad\:OR\:\quad\:5x\:=\:12}$

$\displaystyle{\implies\sf\:x\:=\:4\:\sf\:\quad\:OR\:\quad\:x\:=\:\dfrac{12}{5}}$

But, it is given that x ≠ 4.

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:x\:=\:\dfrac{12}{5}\:}}}}$