Question

Simplify 1/x-3+4/x divided by 4/x-1/x-3

Answer 1

The answer is
`(5x-12)/(3x-12)`

The expression is:
`( (1)/(x-3)+ (4)/(x) )/( (4)/(x) - (1)/(x-3) )`

Let's factorize it:

`( (1)/(x-3)+ (4)/(x) )/( (4)/(x) - (1)/(x-3) ) = ( (x*1)/(x(x-3))+ (4*(x-3))/((x-3)*x))/( ((x-3)*4)/((x-3)*x)- (x*1)/(x*(x-3)))= ( (x+4(x-3))/(x(x-3)) )/( (4(x-3)-x)/(x(x-3)) )`

Since
`( (a)/(b) )/( (c)/(d) )= (a)/(b)/ (c)/(d)= (a*d)/(b*c)`, then:

`( (x+4(x-3))/(x(x-3)) )/( (4(x-3)-x)/(x(x-3)) )= ((x+4(x-3))*x(x-3))/((4(x-3)-x)*x(x-3))`

Cancel x(x-3) and simplify:

`((x+4(x-3))*x(x-3))/((4(x-3)-x)*x(x-3)) = (x+4(x-3))/(4(x-3)-x) = (x+4x-12)/(4x-12 -x) = (5x-12)/(3x-12)`