Let's go through each of these equations and check for those that have no solutions.

1. $3(2x - 2) = 6(x - 1)$

• Left-hand side (LHS): $3(2x - 2) = 6x - 6$
• Right-hand side (RHS): $6(x - 1) = 6x - 6$

Thus, $6x - 6 = 6x - 6$. This equation is true for all values of $x$, meaning it has infinitely many solutions, not no solutions.

2. $4(2x - 3) = 8(x - 2)$

• LHS: $4(2x - 3) = 8x - 12$
• RHS: $8(x - 2) = 8x - 16$

Thus, $8x - 12 = 8x - 16$. Simplifying this: $8x - 8x = -16 + 12$ $0 = -4$

This is a contradiction, so this equation has no solution.

3. $4x - 5 = 5x - 4$

• Simplifying: $4x - 5 = 5x - 4$ $4x - 5x = -4 + 5$ $-x = 1$ $x = -1$

This equation has a solution, $x = -1$.

4. $4x - 3 = 2(3 + 2x)$

• LHS: $4x - 3$
• RHS: $2(3 + 2x) = 6 + 4x$

Thus, $4x - 3 = 4x + 6$. Simplifying: $4x - 4x = 6 + 3$ $0 = 9$

This is a contradiction, so this equation has no solution.

5. $5 + 13x = 5 + 9x$

• Simplifying: $5 + 13x = 5 + 9x$ $13x - 9x = 5 - 5$ $4x = 0$ $x = 0$

This equation has a solution, $x = 0$.

Conclusion:

The equations that have no solutions are:

• $4(2x - 3) = 8(x - 2)$
• $4x - 3 = 2(3 + 2x)$

Would you like more details on these solutions or have further questions?

Here are five related questions to expand on the topic:

1. What does it mean for an equation to have no solution?
2. How can you identify equations with infinitely many solutions?
3. Can systems of equations also have no solutions? What are the conditions for that?
4. How does the distributive property work when simplifying equations?
5. What are the common types of contradictions found in algebraic equations?

Tip: When solving equations, always simplify both sides as much as possible before comparing them—this often reveals whether there are no solutions or infinitely many solutions.