Let's solve the given expression step by step.

The expression is:

\[
5 \frac{1}{2}x + 1 \frac{3}{4} + 2 \left( \frac{3}{4}x - \frac{3}{8} \right)
\]

### Step 1: Convert mixed fractions to improper fractions
1. \( 5 \frac{1}{2}x \) becomes \( \frac{11}{2}x \)
2. \( 1 \frac{3}{4} \) becomes \( \frac{7}{4} \)

The expression now becomes:

\[
\frac{11}{2}x + \frac{7}{4} + 2 \left( \frac{3}{4}x - \frac{3}{8} \right)
\]

### Step 2: Distribute the 2 across the terms in the parentheses
\[
2 \times \frac{3}{4}x = \frac{6}{4}x = \frac{3}{2}x
\]
\[
2 \times \left(-\frac{3}{8}\right) = -\frac{6}{8} = -\frac{3}{4}
\]

Now the expression is:
\[
\frac{11}{2}x + \frac{7}{4} + \frac{3}{2}x - \frac{3}{4}
\]

### Step 3: Combine like terms
First, combine the \( x \)-terms:

\[
\frac{11}{2}x + \frac{3}{2}x = \frac{14}{2}x = 7x
\]

Now, combine the constants:

\[
\frac{7}{4} - \frac{3}{4} = \frac{4}{4} = 1
\]

### Final Expression:
\[
7x + 1
\]

Thus, the simplified form of the expression is:

\[
7x + 1
\]

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Would you like any additional details or have any questions about the steps?

Here are five related questions to expand on this concept:

1. How do you convert mixed numbers into improper fractions?
2. What is the distributive property, and how is it applied in this case?
3. How do you combine like terms involving variables and constants?
4. What are the steps to simplify expressions with parentheses and fractions?
5. Can this method be applied to solve equations with decimals instead of fractions?

**Tip:** Always ensure that terms with variables and constants are combined separately to simplify expressions easily.

Learn how to simplify the algebraic expression 5 1/2x + 1 3/4 + 2(3/4x - 3/8) step by step. This solution covers algebraic concepts like the distributive property, conversion of mixed fractions, and combining like terms. Suitable for students in Grades 7-8.

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