1. **State the problem:** Calculate the value of $2.6 + 0.5 \times \left(1 \frac{3}{4}\right) - \frac{3}{2}$.\n\n2. **Convert mixed number to improper fraction:**\n\n$1 \frac{3}{4} = \frac{4 \times 1 + 3}{4} = \frac{7}{4}$.\n\n3. **Rewrite the expression:**\n\n$2.6 + 0.5 \times \frac{7}{4} - \frac{3}{2}$.\n\n4. **Multiply $0.5$ by $\frac{7}{4}$:**\n\n$0.5 = \frac{1}{2}$, so\n\n$$\frac{1}{2} \times \frac{7}{4} = \frac{7}{8}.$$\n\n5. **Rewrite the expression:**\n\n$2.6 + \frac{7}{8} - \frac{3}{2}$.\n\n6. **Convert decimals and fractions to a common form (fractions):**\n\n$2.6 = \frac{26}{10} = \frac{13}{5}$.\n\n7. **Find common denominator for $\frac{13}{5}$, $\frac{7}{8}$, and $\frac{3}{2}$:**\n\nThe denominators are 5, 8, and 2. The least common denominator (LCD) is 40.\n\n8. **Convert each fraction:**\n\n$\frac{13}{5} = \frac{13 \times 8}{5 \times 8} = \frac{104}{40}$\n\n$\frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40}$\n\n$\frac{3}{2} = \frac{3 \times 20}{2 \times 20} = \frac{60}{40}$.\n\n9. **Rewrite the expression with common denominators:**\n\n$$\frac{104}{40} + \frac{35}{40} - \frac{60}{40}.$$\n\n10. **Combine the fractions:**\n\n$$\frac{104 + 35 - 60}{40} = \frac{79}{40}.$$\n\n11. **Convert the improper fraction to a mixed number or decimal:**\n\n$$\frac{79}{40} = 1 \frac{39}{40} = 1.975.$$\n\n**Final answer:** $1.975$.