Final answer:
To simplify the expression (4x-1)/(2(x-1)) - 3/(2(x-1)(2x-1)), combine the fractions by finding a common denominator of 2(x-1)(2x-1), adjust the terms, and cancel out common factors to get the simplest form 4/(2x-1).
Step-by-step explanation:
To express (4x-1)/(2(x-1)) - 3/(2(x-1)(2x-1)) as a single fraction in its simplest form, we need to find a common denominator and combine the terms. The common denominator here is 2(x-1)(2x-1). We will multiply the numerators appropriately to get equivalent fractions with the same denominator.
Starting with the first fraction, we have (4x-1) which is already over the common denominator of (2(x-1)). For the second fraction, which is -3, this term needs to be adjusted to be over the common denominator. Thus, we multiply -3 by 2 to get -6, which gives us an equivalent fraction. We then have:
(8x-2-6)/(2(x-1)(2x-1))
Simplifying the numerator, we get:
(8x-8)/(2(x-1)(2x-1))
Now, we can see that 8 is a common factor in both terms of the numerator. Factoring 8 out gives us:
8(x-1)/(2(x-1)(2x-1))
We can then cancel out the (x-1) term from the numerator and denominator:
4/(2x-1)
This is the simplest form of the original expression.