Let's solve the equation step by step.

### Given:
\[
-2 + 2x = 6 + 2(x - 4)
\]

1. **Distribute the 2 on the right-hand side**:
\[
-2 + 2x = 6 + 2x - 8
\]

2. **Simplify the right-hand side**:
\[
-2 + 2x = 2x - 2
\]

3. **Subtract \( 2x \) from both sides**:
\[
-2 = -2
\]

This means that the equation holds for all values of \(x\), so the solution is **all real numbers**.

### Conclusion:
The solution is \( x \in \mathbb{R} \), meaning any real number satisfies this equation.

Would you like a more detailed breakdown, or do you have any other questions?

Here are 5 related questions you could explore:
1. What happens if we have different coefficients for \(x\) on both sides?
2. How do we solve quadratic equations with a similar setup?
3. What is the significance of equations that have no solution?
4. What types of systems of equations can result in all real numbers as solutions?
5. How would this equation change if it involved fractions?

**Tip**: Always simplify expressions on both sides of the equation before solving!

Learn how to solve the linear equation -2 + 2x = 6 + 2(x - 4) with a detailed step-by-step solution. This problem demonstrates the distributive property and properties of equality in algebra. Suitable for students in Grades 6-8.

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