https://file.aiquickdraw.com/mathbot/file/17278770572785ro8nugk.jpg

Let's solve each of the four equations and determine which one has no solution.

### Equation 1:
\[
2x + x + 3 = 3(x + 1)
\]
Simplify both sides:
\[
3x + 3 = 3x + 3
\]
This simplifies to:
\[
3x + 3 = 3x + 3
\]
Subtract \(3x\) from both sides:
\[
3 = 3
\]
This is always true, so **Equation 1 has infinitely many solutions**.

### Equation 2:
\[
3(2x + 9) = 6x + 8
\]
Distribute on the left-hand side:
\[
6x + 27 = 6x + 8
\]
Subtract \(6x\) from both sides:
\[
27 = 8
\]
This is a contradiction, so **Equation 2 has no solution**.

### Equation 3:
\[
4x + 5 = 3x + 2x + 3
\]
Simplify both sides:
\[
4x + 5 = 5x + 3
\]
Subtract \(4x\) from both sides:
\[
5 = x + 3
\]
Subtract 3 from both sides:
\[
x = 2
\]
So **Equation 3 has a solution**.

### Equation 4:
\[
6x - 2x + 8 = 4(x + 2)
\]
Simplify both sides:
\[
4x + 8 = 4x + 8
\]
Subtract \(4x\) from both sides:
\[
8 = 8
\]
This is always true, so **Equation 4 has infinitely many solutions**.

### Conclusion:
**Equation 2** has no solution.

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Would you like further details or clarifications?

Here are 5 related questions to explore:

1. How do we identify contradictions in equations?
2. What is the difference between no solution and infinite solutions?
3. Can you provide examples of equations with unique solutions?
4. How does subtracting terms help simplify equations?
5. What is the general process for solving linear equations?

**Tip:** Always check if equations lead to contradictions or identities after simplifying them!

Which equation has no solution? Consider the four equations:
Equation 1: 2x + x + 3 = 3(x + 1)
Equation 2: 3(2x + 9) = 6x + 8
Equation 3: 4x + 5 = 3x + 2x + 3
Equation 4: 6x - 2x + 8 = 4(x + 2)

Learn how to solve a set of four algebraic equations and determine which one has no solution. This problem covers key concepts such as the distributive property, simplifying linear equations, and identifying contradictions in algebraic expressions. Suitable for students in Grades 8-10.

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