To determine which equations have infinitely many solutions, we need to simplify each equation and check if it results in an identity (a true statement for all values of the variable, like $0 = 0$).

Let's analyze each option:

Option A: $3x + x = 4x$

Simplifying: $4x = 4x$ This is an identity since it's true for any value of $x$. Therefore, this equation has infinitely many solutions.

Option B: $3(x + 1) = 3x + 2$

Expanding and simplifying: $3x + 3 = 3x + 2$ Subtracting $3x$ from both sides: $3 = 2$ This is a contradiction, so this equation has no solutions.

Option C: $2(x + 2) = x + 4$

Expanding and simplifying: $2x + 4 = x + 4$ Subtracting $x$ and then subtracting $4$ from both sides: $x = 0$ This equation has a single solution, $x = 0$, not infinitely many.

Option D: $3(x + 1) = 3x + 3$

Expanding and simplifying: $3x + 3 = 3x + 3$ This simplifies to an identity: $0 = 0$ This is true for any value of $x$, so this equation also has infinitely many solutions.

Conclusion

The two equations that have infinitely many solutions are:

• Option A: $3x + x = 4x$
• Option D: $3(x + 1) = 3x + 3$

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. What conditions must be met for a linear equation to have infinitely many solutions?
2. How do you identify an equation with no solutions?
3. Can an equation with one solution ever be an identity? Why or why not?
4. What is the difference between a contradiction and an identity in algebraic equations?
5. How would you verify if a quadratic equation has infinitely many solutions?

Tip: For an equation to have infinitely many solutions, simplifying the equation should result in a statement that is true for any value of the variable, such as $0 = 0$.