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Add: 1/2(x+4)+1/4(4x−2)

. Answer should be in mixed number form.(1 point)

3 answers

To simplify the expression \( \frac{1}{2}(x + 4) + \frac{1}{4}(4x - 2) \), we will first distribute the fractions across their respective expressions.

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

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

  3. Now, combine the two results: \[ \left( \frac{1}{2}x + 2 \right) + \left( x - \frac{1}{2} \right) = \frac{1}{2}x + 2 + x - \frac{1}{2} \]

  4. 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: \[ 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

  5. Putting it all together: \[ \frac{3}{2}x + \frac{3}{2} \]

  6. Factor out the common term, if needed: \[ \frac{3}{2}(x + 1) \]

Since the answer should be represented in mixed number form, \( \frac{3}{2} \) can be expressed as \( 1 \frac{1}{2} \).

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

It was wrong, may you please redo it?

Sure! Let's simplify the expression \( \frac{1}{2}(x + 4) + \frac{1}{4}(4x - 2) \) step by step again.

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

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

  3. Now, combine the two results: \[ \left( \frac{1}{2}x + 2 \right) + \left( x - \frac{1}{2} \right) = \frac{1}{2}x + 2 + x - \frac{1}{2} \]

  4. 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: \[ 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

  5. Putting it all together: \[ \frac{3}{2}x + \frac{3}{2} \]

Now we want to express \( \frac{3}{2} \) in mixed number form.

Converting \( \frac{3}{2} \) to a Mixed Number

\[ \frac{3}{2} = 1 \frac{1}{2} \]

So the expression can be written as: \[ \frac{3}{2}(x + 1) \quad \text{or} \quad 1 \frac{1}{2}x + 1 \frac{1}{2} \]

In summary, the final answer for the expression \( \frac{1}{2}(x + 4) + \frac{1}{4}(4x - 2) \) simplifies to: \[ \frac{3}{2}x + \frac{3}{2} \quad \text{or} \quad 1 \frac{1}{2}x + 1 \frac{1}{2} \]

Please let me know if you need further confirmation or clarification on a specific part.