We are given the equation:

$|2 - 2x| - 6 = 0$

Step 1: Isolate the absolute value

First, isolate the absolute value expression by adding 6 to both sides of the equation:

$|2 - 2x| = 6$

Step 2: Solve for two cases

Since the absolute value of an expression can be either positive or negative, we consider two cases:

Case 1: $2 - 2x = 6$ $2 - 2x = 6$ Subtract 2 from both sides: $-2x = 4$ Divide by -2: $x = -2$

Case 2: $2 - 2x = -6$ $2 - 2x = -6$ Subtract 2 from both sides: $-2x = -8$ Divide by -2: $x = 4$

Step 3: Conclusion

The solutions to the equation are:

$x = -2 \quad \text{or} \quad x = 4$

Would you like more details or have any further questions?

Here are 5 related questions to expand on this:

1. How do we solve absolute value inequalities?
2. What happens when we multiply both sides of an equation by a negative number?
3. How do absolute value functions behave graphically?
4. Can absolute value equations have no solutions? Why or why not?
5. How do you solve systems of equations involving absolute values?

Tip: Always check both cases of an absolute value equation, as it can yield multiple solutions.