Solving rational expressions

Sometimes you are presented with solving for unknown variables in rational expressions (expression where numerator and or denominator are polynomials).

example:

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

First, find least common denominators (lcd). Factor any denominators if possible. We can factor 2x-4 to 2(x-2). This happens to be our lcd. We’ll multiply the first term by 2 and the final term by 2x-4.

(2)*1/(2)*(x-2) = x/(2x-4)+(2x-4)*1/(2x-4)

Now, let’s simplify by multiplying:

2/(2x-4) = x/(2x-4)+(2x-4)/(2x-4)

Let’s add terms on the right side:

2/(2x-4) = (x+2x-4)/(2x-4)

Since the denominators are the same on both sides, the numerators mut be equal:

2 = x+2x-4

Then:

2 = 3x-4

add 4 to both sides:

2+4 = x+2x-4+4

6 = 3x

divide by 3 on both sides:

2 = x

If you plug in 2 to both sides you’ll get undefined. No solutions.