To determine which equation has infinitely many solutions, we need to solve each of the given equations step by step and check for conditions where the equation holds for all values of $x$.

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

Distribute the 3 on the left-hand side: $3x + 3 = 3x + 3$ This simplifies to: $3x + 3x \text{ cancels out, leaving } 3 = 3$ This is a true statement for all values of $x$, meaning this equation has infinitely many solutions.

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

Distribute the 3 on the left-hand side: $3x + 3 = 3$ Simplify: $3x = 0$ Solve for $x$: $x = 0$ This equation has only one solution, $x = 0$.

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

Distribute the 3 on the left-hand side: $3x + 3 = 3x + 1$ Simplify: $3 = 1$ This is a false statement, meaning this equation has no solutions.

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

Distribute the 3 on the left-hand side: $3x + 3 = 1$ Simplify: $3x = -2$ Solve for $x$: $x = -\frac{2}{3}$ This equation has only one solution, $x = -\frac{2}{3}$.

Conclusion:

The equation that has infinitely many solutions is Option A: $3(x + 1) = 3x + 3$.

Would you like more details on any specific part of this explanation?

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Here are 5 related questions:

1. What does it mean for an equation to have no solutions?
2. Can a quadratic equation have infinitely many solutions?
3. How can we identify if a linear equation has exactly one solution?
4. What kind of equation could have no solutions at all?
5. Can a system of equations have infinitely many solutions?

Tip: For an equation to have infinitely many solutions, it must simplify to a true statement (like $0 = 0$) for all values of the variable.