Let's solve each equation to determine which ones have no solution.

A. $x + 6 = 5 + x$

• Subtract $x$ from both sides: $6 = 5$ This is a contradiction, so no solution.

B. $-2(x - 3) = -2x + 6$

• Distribute $-2$ on the left: $-2x + 6 = -2x + 6$ Both sides are identical, so infinite solutions.

C. $4 - 4x = 3x + 2$

• Move terms involving $x$ to one side: $4 - 2 = 3x + 4x$ $2 = 7x$ $x = \frac{2}{7}$ This has a solution: $x = \frac{2}{7}$.

D. $4(x + 1) = 3(x + 2)$

• Distribute both sides: $4x + 4 = 3x + 6$
  • Subtract $3x$ from both sides: $x + 4 = 6$
  • Subtract 4 from both sides: $x = 2$ This has a solution: $x = 2$.

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

• Add $3x$ to both sides: $5 = 4$ This is a contradiction, so no solution.

Conclusion:

The equations with no solution are A and E.

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Would you like further details or explanations on any of the steps?

Here are some related questions:

1. What does it mean when an equation has infinite solutions?
2. How do you recognize when an equation has no solution?
3. Can an equation have both a solution and a contradiction in the same steps?
4. What is the process for solving equations with variables on both sides?
5. How can you check if your solution is correct after solving an equation?

Tip: Always check your final step after simplifying both sides to see if the equation becomes a contradiction (like $6 = 5$) or an identity (like $3x + 5 = 3x + 5$).