Add: \frac{1}{2}(x+4)+\frac{1}{4}(4x-2)
To add the fractions, we first need to find a common denominator. The common denominator for the fractions $\frac{1}{2}$ and $\frac{1}{4}$ is 4.
The expression becomes:
$\frac{1}{2}(x+4)+\frac{1}{4}(4x-2)\\
=\frac{2}{4}(x+4)+\frac{1}{4}(4x-2)\\
=\frac{2(x+4)}{4}+\frac{1(4x-2)}{4}\\
=\frac{2x+8}{4}+\frac{4x-2}{4}$
Now that the fractions have a common denominator, we can combine them:
$\frac{2x+8}{4}+\frac{4x-2}{4}\\
=\frac{(2x+8)+(4x-2)}{4}\\
=\frac{2x+8+4x-2}{4}$
Combining like terms:
$\frac{2x+4x+8-2}{4}\\
=\frac{6x+6}{4}$
To simplify the expression, we can divide both the numerator and the denominator by their greatest common divisor, which is 2:
$\frac{6x+6}{4}\\
=\frac{3(2x+2)}{2(2)}\\
=\frac{3(x+1)}{2}$
So, the simplified expression is $\frac{3(x+1)}{2}$.