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

##### 1 Answer

You have to make the denominators the same, so that you can add the two fractions.

One thing you can do to help determine what the final answer should look like is to graph it. If we set

graph{y = (3x+1)/(x-3) - (4x+1)/(x-2) [-3, 10, -100, 80]}

As you can see, there are (at least) two zeros (or "roots") to this equation. At approximately

First, multiply both fractional parts by a lowest common denominator, namely

With the same denominator, you can now safely add the numerators in a single fraction, like so:

Now, let's use the principle of F.O.I.L. to multiply out the numerator:

Next, we must combine like terms, and simplify:

Many times, at this point, you would try to determine if the numerator factors. Unfortunately, it does not factor into anything neat. In fact, if you apply the quadratic formula to the numerator, you find that

First, near

graph{y = (3x+1)/(x-3) - (4x+1)/(x-2) [-.4, 0, -1, 1]}

And also near

graph{y = (3x+1)/(x-3) - (4x+1)/(x-2) [5.5, 6.5, -1, 1]}.

Therefore,