To solve the expression $2.6 + 0.5 \left( 1 \frac{3}{4} \right) - \frac{3}{2}$, follow these steps:

1. Convert the mixed number to an improper fraction: $1 \frac{3}{4} = 1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}$

2. Simplify the expression inside the parentheses: $0.5 \left( \frac{7}{4} \right) = \frac{1}{2} \left( \frac{7}{4} \right) = \frac{7}{8}$

3. Convert the fractions to a common denominator to subtract: $\frac{7}{8} - \frac{3}{2}$ First, convert $\frac{3}{2}$ to a fraction with a denominator of 8: $\frac{3}{2} = \frac{3 \times 4}{2 \times 4} = \frac{12}{8}$ Now, subtract the fractions: $\frac{7}{8} - \frac{12}{8} = \frac{7 - 12}{8} = \frac{-5}{8}$

4. Combine this result with 2.6: $2.6 + \left( \frac{-5}{8} \right)$

5. Convert 2.6 to a fraction with the same denominator: $2.6 = 2 + 0.6 = 2 + \frac{6}{10} = 2 + \frac{3}{5}$ Convert $\frac{3}{5}$ to a fraction with a denominator of 8: $\frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40}$ Convert 2 to a fraction with a denominator of 40: $2 = \frac{80}{40}$ Now, add these fractions: $2.6 = \frac{80}{40} + \frac{24}{40} = \frac{104}{40}$

6. Combine $\frac{104}{40}$ with $\frac{-5}{8}$ (convert $\frac{-5}{8}$ to a denominator of 40): $\frac{-5}{8} = \frac{-5 \times 5}{8 \times 5} = \frac{-25}{40}$ Now add the fractions: $\frac{104}{40} + \frac{-25}{40} = \frac{104 - 25}{40} = \frac{79}{40}$

So the value of the expression is $\frac{79}{40}$, or 1.975.

However, since this doesn't match the choices exactly, we should check if we missed any option matching our final result. The value 1.975 does not have a straightforward equivalent in the provided options. The best match is the closest value in the context of fraction simplification mistakes. Let's recheck the provided options.

In this case, we observe the calculation seems close to one of the options provided:

From the list:

• $\frac{79}{40} = 1.975$, not an exact match but was closest to 79.98 value.

Reviewing detailed steps seem accurate, provided option does not straightforward align with the detailed steps provided. Consider re-checking step-wise operations in case of matching answers expected.

Do you want any more details or have any questions?

Here are five related questions you might ask:

1. How do you convert between mixed numbers and improper fractions?
2. What are the steps to add and subtract fractions with different denominators?
3. Can you explain how to simplify complex fractions?
4. How do you handle decimal to fraction conversion and vice versa?
5. What is the importance of ensuring a common denominator when working with fractions?

Tip: Always verify final calculations especially when dealing with multiple step operations and comparing against expected choices in answer lists.