To solve the equation \( \frac{11}{12} - \left( \frac{2}{5} + x \right) = \frac{2}{3} \), follow these steps:

1. Simplify the left-hand side of the equation by combining the fractions:
\[
\frac{11}{12} - \frac{2}{5} - x = \frac{2}{3}
\]

2. Find a common denominator for the fractions \(\frac{11}{12}\) and \(\frac{2}{5}\). The least common multiple of 12 and 5 is 60.

Convert \(\frac{11}{12}\) to a fraction with a denominator of 60:
\[
\frac{11}{12} = \frac{11 \times 5}{12 \times 5} = \frac{55}{60}
\]

Convert \(\frac{2}{5}\) to a fraction with a denominator of 60:
\[
\frac{2}{5} = \frac{2 \times 12}{5 \times 12} = \frac{24}{60}
\]

Now the equation is:
\[
\frac{55}{60} - \frac{24}{60} - x = \frac{2}{3}
\]

3. Combine the fractions on the left-hand side:
\[
\frac{55}{60} - \frac{24}{60} = \frac{31}{60}
\]

So the equation becomes:
\[
\frac{31}{60} - x = \frac{2}{3}
\]

4. Isolate \(x\) by subtracting \(\frac{31}{60}\) from both sides:
\[
-x = \frac{2}{3} - \frac{31}{60}
\]

5. Find a common denominator for the fractions \(\frac{2}{3}\) and \(\frac{31}{60}\). The least common multiple of 3 and 60 is 60.

Convert \(\frac{2}{3}\) to a fraction with a denominator of 60:
\[
\frac{2}{3} = \frac{2 \times 20}{3 \times 20} = \frac{40}{60}
\]

Now the equation is:
\[
-x = \frac{40}{60} - \frac{31}{60}
\]

6. Combine the fractions on the right-hand side:
\[
\frac{40}{60} - \frac{31}{60} = \frac{9}{60}
\]

So the equation becomes:
\[
-x = \frac{9}{60}
\]

7. Simplify \(\frac{9}{60}\):
\[
\frac{9}{60} = \frac{3}{20}
\]

So the equation is:
\[
-x = \frac{3}{20}
\]

8. Multiply both sides by -1 to solve for \(x\):
\[
x = -\frac{3}{20}
\]

Therefore, the solution to the equation is:
\[
x = -\frac{3}{20}
\]