Answer
$\dfrac{1}{2x - 1}$
Restrictions: $x \ne 0, \frac{1}{2}$
Work Step by Step
Let's first take a look at the problem to see what we can cancel out from the numerator and denominator:
$\dfrac{2x}{4x^2 - 2x}$
We see that in the denominator, we can factor out $2x$:
$\dfrac{2x}{2x(2x - 1)}$
We can divide the numerator and denominator by $2x$:
$\dfrac{1}{2x - 1}$
To find out if there are any restrictions on the variables, we need to find which values of $x$ and/or $y$ will cause the denominator to equal zero, which would make the whole expression undefined. Let's set the denominator equal to zero and solve for $x$:
$4x^2 - 2x = 0$
Factor out $2x$:
$2x(2x - 1) = 0$
Set each factor equal to zero, according to the zero product property:
First factor:
$2x = 0$
Divide each side by $2$:
$x = 0$
Second factor:
$2x - 1 = 0$
Add $1$ to each side of the equation:
$2x = 1$
Divide each side by $2$:
$x = \frac{1}{2}$
Restriction: $x \ne 0, \frac{1}{2}$
