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

Assuming you want to solve for x, find the common denominator for the left side, #(x+1)(x-4)# and combine them into a single rational equation. Since #4/(x+1)# and #3/(x-4)# combine, it turns into #(7x-13)/((x+1)(x-4))# by multiplying the numerator #4*(x-4)# and #3*(x+1)#, adding them and combining like terms . You want to get rid of any denominators in the problem to make it simpler, so multiply that common denominator to both sides of the equation,
#(7x-13)/((x+1)(x-4))*(x+1)(x+4)# and same with the right side. On the left the common denominator gets cancelled so you're left with #7x-13#. On the right the #(x+1)# gets cancelled and you have #2(x-4)#. Multiply that and overall you have #7x-13=2x-8# left. then bring the #x's# on one side and the rest on the other side. #7x-2x=-8+13# , add and finally you have, #5x=5#, #x=1#. To check you can plug #x# in the original equation.

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

Our strategy will be to eliminate quotients.

This equation will look a lot less scary if we multiply both sides by #x+1#.

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

Now subtract 4 from both sides of the equation.

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

Now multiply both side s of the equation by #x-4#.

#3(x+1)=-2(x-4)#

Hey! No more quotients! Now apply the distributive property.

#3x+3=-2x+8#

Add #2x# to both sides of this equation.

#5x+3=8#

Subtract 3 from both sides of this equation.

#5x=5#

Divide both sides of this equation by 5.

#x=1#

We can check our answer.

#4/(1+1)+3/(1-4)# =?= #2/(1+1)#

#4/2+3/-3# =?= #2/2#

#2-1# =?= #1#

#1=1#

Wadya know? Algebra WORKS!