Add: 1/2 ( x + 4 ) + 1/4 ( 4 x − 2 )

To add the expressions \( \frac{1}{2} ( x + 4 ) + \frac{1}{4} ( 4x - 2 ) \), we first simplify each term separately.

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

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

Now, we can add the two results together: \[ \left( \frac{1}{2} x + 2 \right) + \left( x - \frac{1}{2} \right) \]

Combine like terms:

• The \( x \) terms: \[ \frac{1}{2} x + x = \frac{1}{2} x + \frac{2}{2} x = \frac{3}{2} x \]

• The constant terms: \[ 2 - \frac{1}{2} = 2 - 0.5 = 1.5 = \frac{3}{2} \]

Therefore, the combined result is: \[ \frac{3}{2} x + \frac{3}{2} \]

To express this in mixed number form, we can factor out \( \frac{3}{2} \): \[ \frac{3}{2} ( x + 1 ) \]

Thus, the final answer is: \[ \frac{3}{2} ( x + 1 ) \]

Apologies for any confusion in my previous response. Let's go through the problem step-by-step again and find the correct answer.

We want to simplify the expression:

\[ \frac{1}{2} (x + 4) + \frac{1}{4} (4x - 2) \]

1. Distribute the terms in the parentheses:

   • For \( \frac{1}{2}(x + 4) \): \[ \frac{1}{2}x + \frac{1}{2} \cdot 4 = \frac{1}{2}x + 2 \]
   • For \( \frac{1}{4}(4x - 2) \): \[ \frac{1}{4} \cdot 4x - \frac{1}{4} \cdot 2 = x - \frac{1}{2} \]

2. Combine the results: \[ \left(\frac{1}{2}x + 2\right) + \left(x - \frac{1}{2}\right) \]

3. Combine like terms:

   • Combine the \(x\) terms: \[ \frac{1}{2}x + x = \frac{1}{2}x + \frac{2}{2}x = \frac{3}{2}x \]
   • Combine the constant terms: \[ 2 - \frac{1}{2} = 2 - 0.5 = 1.5 = \frac{3}{2} \]

4. Final expression: Now we have: \[ \frac{3}{2}x + \frac{3}{2} \]

5. Expressing in mixed number form: This expression can be factored if necessary, but in mixed number form for the term \( \frac{3}{2} \), you can express it as \( 1 \frac{1}{2} \). So the entire expression can be written as: \[ \frac{3}{2}(x + 1) \]

However, if you wish to give just the constant part in mixed number format, the result is: \[ \frac{3}{2}x + 1 \frac{1}{2} \]

If there was any specific part of the question I misunderstood or any specific aspect of the mixed number format you wanted, please let me know!