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

Question

Avery Alexander · Accepted Answer

#(x+3)/(x+2)# #"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#

Nathan Axel · Answer

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

Jace Anderson · Answer

undefined To simplify the expression (2x-1)/((x-3)(x+2)) + (x-4)/(x-3), we need to find a common denominator and combine the fractions. The common denominator is (x-3)(x+2).

To add the fractions, we multiply the numerator of the first fraction (2x-1) by (x+2) and the numerator of the second fraction (x-4) by (x-3).

Expanding these expressions, we get (2x^2 + 3x - 2) + (x^2 - 7x + 12).

Combining like terms, we have 3x^2 - 4x + 10.

Therefore, the simplified expression is (3x^2 - 4x + 10)/((x-3)(x+2)).