Algebra question

Question

How do I simplify

x/x+1 = 1/2x-1 +2/x-1

Answer 1

So we're asked to "simplify" the equation:

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

or

x/(x+1) - 1/(2x-1) - 2/(x-1) = 0. (Equation A)

So we have 3 fractions, all of which have a different denominator.

(And note that the function is undefined at x = -1, 1/2, and 1, since that would cause a divide-by-zero 'error'.)

In this situation, "simplify" usually includes giving the fractions a common denominator.

So we'll multiply each fraction by 1, in a special form that will give us our common denominator:

[x/(x+1)] * [(2x-1)*(x-1) / (2x-1)*(x-1)]

-[1/(2x-1] * [(x+1)*(x-1) / (x+1)*(x-1)]

-[2/(x-1)] * [(x+1)*(2x-1) / (x+1)*(2x-1)] = 0

After multiplying out the numerators and collecting like terms, we eventually get

(2x³ -8x² -x +3) / [(x+1)*(2x-1)*(x-1)] = 0

Is this form "more simple" than the initial problem statement? It's debatable! ;-)

Perhaps the easiest way to check our work is to graph the function and compare it with the graph of the original function. They should match.

Answer 2

Multiple all three terms by the common denominator:

(x + 1)(2x -1)(x - 1)