Question

Add: 1/2 ( x + 4 ) + 1/4 ( 4 x − 2 ). Answer should be in mixed number form.(1 point)

Answer 1

To simplify the expression \( \frac{1}{2} ( x + 4 ) + \frac{1}{4} ( 4x - 2 ) \), we will first distribute the fractions:

1. Distributing \( \frac{1}{2} \) to \( ( x + 4 ) \):
\[
\frac{1}{2}(x + 4) = \frac{1}{2}x + \frac{1}{2} \cdot 4 = \frac{1}{2}x + 2
\]

2. Distributing \( \frac{1}{4} \) to \( ( 4x - 2 ) \):
\[
\frac{1}{4}(4x - 2) = \frac{1}{4} \cdot 4x - \frac{1}{4} \cdot 2 = x - \frac{1}{2}
\]

Now, we can combine the results:
\[
\frac{1}{2}x + 2 + x - \frac{1}{2}
\]

Next, we can combine like terms:
- For the \( x \) terms:
\[
\frac{1}{2}x + x = \frac{1}{2}x + \frac{2}{2}x = \frac{3}{2}x
\]

- For the constant terms:
\[
2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}
\]

Thus, we can combine everything together:
\[
\frac{3}{2}x + \frac{3}{2}
\]

To express this in mixed number form, we observe that both coefficients can be factored:
\[
\frac{3}{2}(x + 1)
\]

Now, in this instance, the constants do not need further conversion since they are already proper fractions.

Therefore, the final answer in mixed number form for the expression \( \frac{3}{2}x + \frac{3}{2} \) is:
\[
\frac{3}{2}(x + 1) \text{ or simply } \frac{3}{2}x + \frac{3}{2}
\]
(you could express this as \( 1 \frac{1}{2} x + 1 \frac{1}{2} \) if you prefer the mixed number format without factoring as well).