We are given the equation $\lvert 2x - 6 \rvert = 8$.

To solve this equation, we need to break it down into two cases because of the absolute value:

Case 1: $2x - 6 = 8$

Solve for $x$: $2x - 6 = 8$ Add 6 to both sides: $2x = 14$ Divide by 2: $x = 7$

Case 2: $2x - 6 = -8$

Solve for $x$: $2x - 6 = -8$ Add 6 to both sides: $2x = -2$ Divide by 2: $x = -1$

Thus, the two solutions are $x = 7$ and $x = -1$.

The correct answers are $7$ and $-1$.

Let me know if you would like more details or have any further questions! Here are 5 related questions:

1. What are absolute value equations and how are they generally solved?
2. Can you apply absolute value principles to inequalities?
3. What happens when the absolute value equation has no solution?
4. How can you verify the solutions of an absolute value equation?
5. What are common mistakes to avoid when solving absolute value equations?

Tip: Always remember to check both the positive and negative cases when solving absolute value equations!