Candice wrote the four equations below. She examined them, without solving them, to determine which equation has an infinite number of solutions.

Question

Negative 5 x + 1 = negative 5 x + 1. Negative 2 x + 1 = negative 2 x + 2. 3 x + 5 = 3 x minus 5. 4 x minus 2 = x + 1.

Which of Candice’s equations has an infinite number of solutions?
Negative 5 x + 1 = negative 5 x + 1
Negative 2 x + 1 = negative 2 x + 2
3 x + 5 = 3 x minus 5
4 x minus 2 = x + 1

Answer 1

To determine which of Candice's equations has an infinite number of solutions, we need to analyze each equation:

1. Negative 5x + 1 = negative 5x + 1
   This equation states that both sides are identical, so it is true for all values of x. Therefore, it has an infinite number of solutions.

2. Negative 2x + 1 = negative 2x + 2
   If we simplify this equation:
   \(-2x + 1 - (-2x) = 2\)
   This simplifies to \(1 = 2\), which is false. Therefore, it has no solutions.

3. 3x + 5 = 3x - 5
   If we simplify this equation:
   \(3x + 5 - 3x = -5\)
   This simplifies to \(5 = -5\), which is also false. Therefore, it has no solutions.

4. 4x - 2 = x + 1
   If we simplify this equation:
   \(4x - x = 1 + 2\)
   This simplifies to \(3x = 3\), or \(x = 1\). Therefore, it has exactly one solution.

Based on this analysis, the equation that has an infinite number of solutions is:

Negative 5x + 1 = negative 5x + 1.