Question

How do you simplify `(2x-1)/((x-3)(x+2)) + (x-4)/(x-3)`?

Answer 1

`(x+3)/(x+2)`

Explanation:

`"We require the fractions to have common denominators"`

`"multiply the numerator/denominator of"`

`(x-4)/(x-3)" by "(x+2)`

`=(2x-1)/((x-3)(x+2))+((x-4)(x+2))/((x-3)(x+2))`

`=(2x-1)/((x-3)(x+2))+(x^2-2x-8)/((x-3)(x+2))`

`"add terms on numerators leaving the denominator"`

`=(x^2-9)/((x-3)(x+2))larrcolor(blue)"difference of squares"`

`=(cancel((x-3))(x+3))/(cancel((x-3))(x+2))larrcolor(blue)"cancel common factor"`

`=(x+3)/(x+2)`

`"with restriction "x!=3`

Answer 2

`color(brown)(=> (x+ 3) / (x+2)`

Explanation:

`(2x-1)/((x-3)(x+2)) + (x-4)/(x-3)`

`=>( (2x-1) + ((x-4)(x+2))) / ((x-3)(x+2))`, taking L C M.

`=> (cancel(2x) - 1 + x^2 - cancel(2x) - 8) / ((x-3)(x+2))`

`=> (x^2 - 9) / ((x-3)(x+2))`

`=> ((x+3)cancel(x-3)) / (cancel(x-3)(x+2))`

`color(brown)(=> (x+ 3) / (x+2)`