The equation in the image reads as follows:

$\frac{1}{2} - 1 - \left[ \left( \frac{2}{3} - \frac{1}{2} \right) + \frac{3}{5} \cdot \left( -\frac{5}{8} \right) \right] + \frac{1}{2}.$

Step-by-Step Solution:

1. Simplify the parentheses:

   Inside the brackets: $\frac{2}{3} - \frac{1}{2}.$ Find the common denominator (6): $\frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6}.$ Subtract: $\frac{4}{6} - \frac{3}{6} = \frac{1}{6}.$

   Now, the expression becomes: $\frac{1}{2} - 1 - \left[ \frac{1}{6} + \frac{3}{5} \cdot \left( -\frac{5}{8} \right) \right] + \frac{1}{2}.$

2. Simplify the product inside the brackets:

   Multiply: $\frac{3}{5} \cdot \left( -\frac{5}{8} \right).$ Simplify: $= -\frac{15}{40}.$ Reduce: $= -\frac{3}{8}.$

   Substitute back into the expression: $\frac{1}{2} - 1 - \left[ \frac{1}{6} - \frac{3}{8} \right] + \frac{1}{2}.$

3. Simplify the terms in the brackets:

   Find a common denominator for $\frac{1}{6}$ and $-\frac{3}{8}$ (24): $\frac{1}{6} = \frac{4}{24}, \quad -\frac{3}{8} = -\frac{9}{24}.$ Add: $\frac{4}{24} - \frac{9}{24} = -\frac{5}{24}.$

   Now, the expression is: $\frac{1}{2} - 1 - \left( -\frac{5}{24} \right) + \frac{1}{2}.$

4. Simplify further:

   Remove the brackets: $\frac{1}{2} - 1 + \frac{5}{24} + \frac{1}{2}.$

   Combine $\frac{1}{2}$ and $\frac{1}{2}$: $1 - 1 + \frac{5}{24}.$

5. Final simplification:

   $1 - 1 + \frac{5}{24} = \frac{5}{24}.$

Final Answer:

$\boxed{\frac{5}{24}}$

Would you like a deeper explanation for any step, or do you have any questions?

Related Questions:

1. How can you find a common denominator when simplifying fractions?
2. What are the steps for distributing multiplication over addition in fractions?
3. How does the order of operations work with nested brackets?
4. What are the general methods for reducing fractions to their simplest form?
5. How does multiplying two fractions result in simplification?

Tip:

Always handle brackets and parentheses first, simplifying step-by-step to avoid errors.