simplify [(x+3)/(2x^2+9x+9)]+ [1/2(2x-3)]-[4x/(4x^2-9)]

Question

Simplifying of rational numbers by using rules of addition of rational numbers

Answer 1

2x^2+9x+9=(2x+3)(x+3) so (x+3)/(2x^2+9x+9)=(x+3)/((2x+3)(x+3))=1/(2x+3) (as long as x≠-3).

4x^2-9=(2x+3)(2x-3), 1/(2(2x-3))-4x/(4x^2-9)=1/(2(2x-3))-4x/((2x+3)(2x-3)).

We can single out the common factor 1/(2x-3): (1/(2x-3))(1/2-4x/(2x+3)).

This becomes: (1/(2x-3))(2x+3-8x)/(2(2x+3))=(3/(2x-3))(1-2x)/(2(2x+3)).

Combining the two components:

1/(2x+3)+(3/2)(1-2x)/((2x-3)(2x+3)).

We can single out the factor 1/(2x+3):

(1/(2x+3))(1+(3/2)(1-2x)/(2x-3))=(1/(2x+3))(4x-6+3-6x)/(2(2x-3)).

-(1/(2x+3))(2x+3)/(2(2x-3))=-1/(2(2x-3)).