Final answer:
After applying the distributive property and simplifying the equation, we find that both sides are identical, leading to the conclusion that there are infinitely many solutions to the given equation.
Step-by-step explanation:
How many solutions exist for the given equation 12x + 1 = 3(4x + 1) - 2? To determine the number of solutions, we need to simplify and solve the equation for x.
First, we apply the distributive property to the right side of the equation:
12x + 1 = 3(4x) + 3(1) - 2
12x + 1 = 12x + 3 - 2
When we simplify further, we combine like terms:
12x + 1 = 12x + 1
At this point, we notice that both sides of the equation are identical, which means any value for x will satisfy the equation. Therefore, this is a case of infinitely many solutions.