Solve {x}/{|x|} + {y}/{|y|} + {z}/{|z|} + {xyz}/{|xyz|} = 0.

Never mind, I have figured it out.

To solve the given equation, let's analyze each term separately and then combine them.

1. {x}/{|x|}:
The expression {x}/{|x|} represents the sign of x. When x is positive, {x}/{|x|} = 1, and when x is negative, {x}/{|x|} = -1. Since we are summing this term with the others, this means we need to consider the sign of x for both positive and negative values.

2. {y}/{|y|}:
Similar to the first term, this represents the sign of y. When y is positive, {y}/{|y|} = 1, and when y is negative, {y}/{|y|} = -1.

3. {z}/{|z|}:
Again, this represents the sign of z. When z is positive, {z}/{|z|} = 1, and when z is negative, {z}/{|z|} = -1.

4. {xyz}/{|xyz|}:
This term is slightly different. When xyz is positive, {xyz}/{|xyz|} = 1, and when xyz is negative, {xyz}/{|xyz|} = -1. However, if any of x, y, z is zero, then xyz is also zero, and the equation becomes indeterminate. So, we need to exclude that case.

Now, let's consider possibilities for the sum to equal zero. There are a few cases to explore:

Case 1: All terms are positive (x > 0, y > 0, z > 0)
In this case, all the terms will be 1. So, 1 + 1 + 1 + 1 = 4, which is not equal to zero. Therefore, this case doesn't satisfy the equation.

Case 2: Two terms are positive and one is negative
Without loss of generality, let's assume x > 0, y < 0, and z > 0.
{x}/{|x|} = 1
{y}/{|y|} = -1
{z}/{|z|} = 1
{xyz}/{|xyz|} = -1 (since xy is negative)
So, 1 - 1 + 1 - 1 = 0.

Case 3: Two terms are negative and one is positive
Without loss of generality, let's assume x < 0, y < 0, and z > 0.
{x}/{|x|} = -1
{y}/{|y|} = -1
{z}/{|z|} = 1
{xyz}/{|xyz|} = 1 (since xyz is positive)
So, -1 - 1 + 1 + 1 = 0.

Case 4: One term is positive and two are negative
Without loss of generality, let's assume x > 0, y < 0, and z < 0.
{x}/{|x|} = 1
{y}/{|y|} = -1
{z}/{|z|} = -1
{xyz}/{|xyz|} = -1 (since xyz is negative)
So, 1 - 1 - 1 - 1 = -2, which is not equal to zero.

Therefore, the only solutions to the equation are when two terms are positive and one is negative, or when two terms are negative and one is positive.