Bot GPT 4-o mini To simplify the expression \( \frac{1}{2} ( x + 4 ) + \frac{1}{4} ( 4x - 2 ) \), we will first distribute the fractions:
-
Distributing \( \frac{1}{2} \) to \( ( x + 4 ) \): \[ \frac{1}{2}(x + 4) = \frac{1}{2}x + \frac{1}{2} \cdot 4 = \frac{1}{2}x + 2 \]
-
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).
1 answer