Need help simplifying?

Question

Simplify: (1/2x)+(2/x-1)

The answer is (5x-1)/(2x^2-2x)

Please show how to work this out

Answer 1

LCD = 2x(x-1)

the first function they had 2x so you only need to time the top by (x-1), second function is (x-1) so you have to time that with 2x to have the same LCD. or you can do crossed multiply.
(1*(x-1)+2*2x)/(2x*(x-1) distribute the bottom

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

(5x-1)/(2x^2-2x)

Answer 2

This is a case of finding the least common denominator, just as you did with ratios of integers (aka fractions.) With polynomials, you have a whole new selection of factors. In this case, you have the factors 2 and x in the denominator of the first fraction and the binomial x - 1 in the denominator of your second. Your least common denominator is therefore 2*x*(x - 1)

Once you have your least common denominator, you can then divide the least common denominator by the existing denominator of each term and multiply the term by the quotient. Or you can take the alternate approach of multiplying by the missing factors, which amounts to the same thing.

In 1/(2x), the missing factor in the denominator is x - 1, so you multiply both numerator and denominator by x - 1. Similarly, the denominator of 2 / (x - 1) is missing both the 2 and the x, so you multiply both numerator and denominator by 2x. This gives

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

Now that the denominators are the same, it is possible to add the numerators and use the common denominator:
(x-1)/[2x(x-1)] + 4x/[2x(x-1)]
= (x - 1 + 4x) / [2x (x - 1)]
= (5x - 1) / [2x (x - 1)]

Before simplifying the denominator, it is a good idea to look for common factors in the numerator and denominator. There are none here. Nevertheless, you must simplify these fractions just as you simplified the simpler ones consisting only of integers. It is sometimes necessary to factor the numerator to do this, but in this problem the numerator cannot be factored: it is already prime.

Seeing that there are no common factors, it is a matter of simplifying the denominator, giving (5x - 1) / (2x² - 2x). In practice, the answer is often left in factored form, especially if this is a partial result that will be an operand for further mathematical manipulation.

By the way, it may look as if Jun Agruda is wrong, and he is - sort of - to interpret the problem as 1/(2x) + 2/x - 1. Instead, he has interpreted the problem more or less as you wrote it, supplying the missing parentheses in the denominator of the first fraction but not the second. When transcribing rational expressions into one line, you must remember to include the parentheses implied by the long divisor bar.

Answer 3

1/2x + 2/x - 1
The least common denominator of 2x and x is 2x:

1/2x + 4/2x - 1 = 0
5/2x = 1
5 = 2x
2 1/2 = x

Answer: x = 2 1/2

Proof (substitute x with 2 1/2):
(1/[2 * 2 1/2]) + (2/[2.5]) - 1 = 0
(1/5) + (2/2.5) - 1 = 0
0.2 + 0.8 - 1 = 0
1 - 1 = 0
0 = 0

Answer 4

Have guessed question to be:-
1 / (2x) + 2 / (x - 1)
[ (x - 1) + 4x ] / [ (2x) (x - 1) ]
(5x - 1) / [ (2x) (x - 1) ]