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The problem in the image asks to write an algebraic expression as a sum. The expression is:

\[
\frac{1}{4}x + 1 + \frac{3}{4}x - \frac{2}{3} - \frac{1}{2}x
\]

To write this as a sum of two terms, we can combine like terms and simplify the expression.

### Step 1: Combine like terms involving \(x\)
The terms involving \(x\) are:

\[
\frac{1}{4}x + \frac{3}{4}x - \frac{1}{2}x
\]

First, convert all terms to have the same denominator. The denominator of the first two terms is 4, while the third term has 2 as the denominator. So, we rewrite \(-\frac{1}{2}x\) as \(-\frac{2}{4}x\) to match the denominator:

\[
\frac{1}{4}x + \frac{3}{4}x - \frac{2}{4}x
\]

Now, add the coefficients of \(x\):

\[
\left(\frac{1}{4} + \frac{3}{4} - \frac{2}{4}\right)x = \frac{2}{4}x = \frac{1}{2}x
\]

### Step 2: Combine the constants
The constant terms are \(1\) and \(-\frac{2}{3}\). To add these, we need to express them with a common denominator. The common denominator between 1 and \(\frac{2}{3}\) is 3, so we rewrite \(1\) as \(\frac{3}{3}\):

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

### Final Expression
Now, the simplified expression is:

\[
\frac{1}{2}x + \frac{1}{3}
\]

So the expression written as a sum is:

\[
\boxed{\frac{1}{2}x + \frac{1}{3}}
\]

Would you like further details or clarifications on this process?

Here are some related questions for you:
1. How do you find a common denominator when adding fractions with different denominators?
2. What is the process of combining like terms in algebraic expressions?
3. How do you simplify an expression involving both constants and variables?
4. How would the solution change if there were additional \(x\) terms or constants?
5. Can you explain why it's important to rewrite fractions with a common denominator when adding them?

**Tip:** Always check that you've combined all like terms and simplified fractions to their lowest terms to avoid mistakes in algebraic expressions.

First write the expression \(\frac{1}{4}x + 1 + \frac{3}{4}x - \frac{2}{3} - \frac{1}{2}x\) as a sum.

Learn how to simplify the algebraic expression \(\frac{1}{4}x + 1 + \frac{3}{4}x - \frac{2}{3} - \frac{1}{2}x\) by combining like terms and adding fractions. This solution includes step-by-step guidance for students in Grades 6-8.

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