Let's get a common denominator for the LHS! Just like if you had 2/3+1/5, you would have an LCD of 3*5=15, let's take the same approach. Here our LCD is (x+1)(x+4). So to make the denominator of each (x+1)(x+4), I think of it as multiplying both terms on the left by (x+1)(x+4)/(x+1)(x+4). So x/(x+1) becomes x(x+4)/(x+1)(x+4) and 3/(x+4) becomes 3(x+1)/(x+1)(x+4). Now we can combine the like terms on the LHS since the denominator is the same. So we have (x*(x+4)+3*(x+1))/(x+1)(x+4)=(x+3)/(x+4). Now we can multiply both sides by x+4 since we see we can get some nice cancellation. We have (x*(x+4)+3*(x+1)/(x+1)=x+3. Now multiply both sides by (x+1), expand, and have a field day: (x^2+4x+3x+3)=x^2+4x+3 This simplifies to 3x=0. So x=0. We plug in, and see that it is indeed correct. Wolfram Alpha verifies it: http://www.wolframalpha.com/input/?i=(x/(x+1))+3/(x+4)=(x+3)/(x+4) Does that make sense? Let me know if something is unclear.

I was thinking a little differently: You could subtract the 3/(x+4) to the other side of the equation, so you have: x/(x+1) = (x+3)/(x+4) - 3/(x+4) Then combine the two fractions on the right: x/(x+1) = (x+3-3)/(x+4) Then I would simplify: x/(x+1) =x/(x+4) Cross multiply: x(x+4) = x(x+1) Simplify and solve: x^2 + 4x = x^2 + x 3x=0, so x=0.

That is perfectly clear, Mathemagician. Thank you so much. I guess I just needed a magician to clear things up. I had somehow messed up the denominators along the way... I appreciate your help.

That is really interesting, mopar. Thank you for that insight. Did you jump to that solution right away because of the x + 4 denominator on both sides? I was so fixated on getting a LCD that I didn't even think of that.

As soon as I saw the x+4 on both sides, I thought it would be much easier than trying to deal with multiply by (x+1)(x+4) to three fractions.

Np, mopar has a much more efficient solution actually! Sometimes with math you can't see the forest through the trees. When I see something like this, I am so quick to find a common denominator that easy tricks slip by me. At least we know both methods work haha!

mopar...I would do this.. x/(x + 1) = x/(x + 4) from here x=0 is clearly a solution so now we can cancel the x's 1/(x + 1) = 1/(x + 4) reciprocate x+1=x+4 1=4 thus there an no other solutions. ie the only solution is x=0