The problem asks to simplify the expression $1\frac{1}{2} + 2\frac{1}{3} \times \frac{3}{4} - \frac{1}{2}$.

1. Problem Description

The problem asks to simplify the expression $1\frac{1}{2} + 2\frac{1}{3} \times \frac{3}{4} - \frac{1}{2}$.

2. Solution Steps

First, we convert the mixed numbers to improper fractions:

$1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}$

$2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$

Now we can rewrite the expression as:

$\frac{3}{2} + \frac{7}{3} \times \frac{3}{4} - \frac{1}{2}$

Next, we perform the multiplication:

$\frac{7}{3} \times \frac{3}{4} = \frac{7 \times 3}{3 \times 4} = \frac{21}{12} = \frac{7}{4}$

So the expression becomes:

$\frac{3}{2} + \frac{7}{4} - \frac{1}{2}$

Now, we find a common denominator, which is 4:

$\frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4}$

$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

The expression becomes:

$\frac{6}{4} + \frac{7}{4} - \frac{2}{4}$

Now we perform the addition and subtraction from left to right:

$\frac{6}{4} + \frac{7}{4} = \frac{6 + 7}{4} = \frac{13}{4}$

$\frac{13}{4} - \frac{2}{4} = \frac{13 - 2}{4} = \frac{11}{4}$

Finally, we convert the improper fraction $\frac{11}{4}$ to a mixed number:

$\frac{11}{4} = 2\frac{3}{4}$

3. Final Answer

$2\frac{3}{4}$