A) To solve the equation -7/3x - 19/6 = 1 - 1/2, we can start by simplifying both sides of the equation.

First, let's simplify the left side of the equation:
-7/3x - 19/6 = -14/6x - 19/6

Now, let's simplify the right side of the equation:
1 - 1/2 = 2/2 - 1/2 = 1/2

So the equation becomes:
-14/6x - 19/6 = 1/2

To get rid of the fractions, we can multiply both sides of the equation by 6:
6 * (-14/6x - 19/6) = 6 * (1/2)
-14x - 19 = 3

Next, let's isolate the variable x by moving the constant term to the other side of the equation:
-14x = 3 + 19
-14x = 22

Finally, divide both sides of the equation by -14 to solve for x:
x = 22 / -14
x = -11/7

Therefore, the solution to the equation -7/3x - 19/6 = 1 - 1/2 is x = -11/7.

B) To solve the equation 3/2x + 2/3 = 4 - 4 - 5/6, we can start by simplifying both sides of the equation.

First, let's simplify the left side of the equation:
3/2x + 2/3 = 9/6x + 2/3

Now, let's simplify the right side of the equation:
4 - 4 - 5/6 = 0 - 5/6 = -5/6

So the equation becomes:
9/6x + 2/3 = -5/6

To get rid of the fractions, we can multiply both sides of the equation by 6:
6 * (9/6x + 2/3) = 6 * (-5/6)
9x + 4 = -5

Next, let's isolate the variable x by moving the constant term to the other side of the equation:
9x = -5 - 4
9x = -9

Finally, divide both sides of the equation by 9 to solve for x:
x = -9 / 9
x = -1

Therefore, the solution to the equation 3/2x + 2/3 = 4 - 4 - 5/6 is x = -1.

C) To solve the equation -3x + 1/2 = 2 - 4/5, we can start by simplifying both sides of the equation.

First, let's simplify the right side of the equation:
2 - 4/5 = 10/5 - 4/5 = 6/5

So the equation becomes:
-3x + 1/2 = 6/5

To get rid of the fraction, we can multiply both sides of the equation by 2:
2 * (-3x + 1/2) = 2 * (6/5)
-6x + 1 = 12/5

Next, let's isolate the variable x by moving the constant term to the other side of the equation:
-6x = 12/5 - 1
-6x = 12/5 - 5/5
-6x = 7/5

Finally, divide both sides of the equation by -6 to solve for x:
x = (7/5) / -6
x = 7/5 * -1/6
x = -7/30

Therefore, the solution to the equation -3x + 1/2 = 2 - 4/5 is x = -7/30.

D) To solve the equation -5/6 + 3x = 2/3 - 1/x, we can start by simplifying both sides of the equation.

First, let's simplify the left side of the equation:
-5/6 + 3x = -5/6 + 18x/6

Now, let's simplify the right side of the equation:
2/3 - 1/x = 2/3 - 1 * 6/x = 2/3 - 6/x

So the equation becomes:
-5/6 + 18x/6 = 2/3 - 6/x

To get rid of the fractions, we can multiply both sides of the equation by 6x:
6x * (-5/6 + 18x/6) = 6x * (2/3 - 6/x)
-5x + 18x^2 = 12x - 36

Next, let's move all the terms to one side of the equation:
18x^2 - 5x - 12x + 36 = 0
18x^2 - 17x + 36 = 0

Unfortunately, this equation cannot be easily factored, so we can use the quadratic formula to solve for x:
x = (-(-17) ± √((-17)^2 - 4(18)(36))) / (2(18))
x = (17 ± √(289 - 2592)) / 36
x = (17 ± √(-2303)) / 36

Since the square root of a negative number is not a real number, there are no real solutions to this equation.

Therefore, the equation -5/6 + 3x = 2/3 - 1/x has no real solutions.