Let’s solve this step by step, simplifying both expressions and comparing them.

Expression 1:

$\frac{-13 + (-2)^2 - ( - | -7 | )}{3 - 2 \times 6} \times \frac{3(2)^{-1}}{2(-3)^{-1}}$

1. Simplify the numerator of the first fraction:

   • $(-2)^2 = 4$,
   • $- | -7 | = -7$ (the absolute value of $-7$ is $7$),
   • So, the numerator becomes: $-13 + 4 - (-7) = -13 + 4 + 7 = -2.$

2. Simplify the denominator of the first fraction:

   • $3 - 2 \times 6 = 3 - 12 = -9.$

   So, the first fraction is: $\frac{-2}{-9} = \frac{2}{9}.$

3. Simplify the second fraction:

   • $3(2)^{-1} = 3 \times \frac{1}{2} = \frac{3}{2},$
   • $2(-3)^{-1} = 2 \times -\frac{1}{3} = -\frac{2}{3}.$
   • So, the second fraction is: $\frac{\frac{3}{2}}{-\frac{2}{3}} = \frac{3}{2} \times \frac{3}{-2} = \frac{9}{-4} = -\frac{9}{4}.$

4. Multiply the two fractions: $\frac{2}{9} \times -\frac{9}{4} = \frac{-18}{36} = -\frac{1}{2}.$

Expression 2:

$\frac{-\left(2^2 \times 3^2 + 2\right) + 4\left[6 - (-2)^3\right]^3}{-4 + 2(-2)} + 1 \frac{3}{4}$

1. Simplify the numerator:

   • $2^2 = 4,$ $3^2 = 9,$ so $2^2 \times 3^2 = 4 \times 9 = 36.$
   • Add $2$: $36 + 2 = 38.$
   • $-\left(2^2 \times 3^2 + 2\right) = -38.$

   For $4\left[6 - (-2)^3\right]^3$:

   • $(-2)^3 = -8,$ so $6 - (-2)^3 = 6 - (-8) = 6 + 8 = 14.$
   • $14^3 = 14 \times 14 \times 14 = 2744,$ so $4[6 - (-2)^3]^3 = 4 \times 2744 = 10976.$

   So, the numerator becomes: $-38 + 10976 = 10938.$

2. Simplify the denominator:

   • $-4 + 2(-2) = -4 - 4 = -8.$

   So, the fraction is: $\frac{10938}{-8} = -1367.25.$

3. Add $1 \frac{3}{4} = 1.75$: $-1367.25 + 1.75 = -1365.5.$

Comparison:

• Expression 1: $-\frac{1}{2} = -0.5.$
• Expression 2: $-1365.5.$

Since: $-0.5 > -1365.5,$ the appropriate symbol is: $> .$

Final Answer:

$\boxed{>}$

Would you like me to explain any specific steps or go further into any part of the solution? Here are some related questions to explore:

1. How do you simplify expressions with absolute values and exponents?
2. What are the rules for multiplying and dividing fractions?
3. How do you compute fractional exponents, such as $(2)^{-1}$?
4. Why does the denominator determine the direction of inequality in fractions?
5. What is the order of operations for nested brackets?

Tip: Always evaluate expressions inside parentheses or brackets first, following the order of operations (PEMDAS).