Noah solved an equation as shown below and found that the equation has infinitely many solutions.

Question

Negative 3 (x + 4) + 2 x = 2 (x minus 6) minus 3 x. Negative 3 x minus 12 + 2 x = 2 x minus 12 minus 3 x. Negative x minus 12 = negative x minus 12.

Which explains whether Noah is correct?
Noah is correct because the two sides of the equation are equivalent expressions.
Noah is correct because if he continues the solution, the final solution will be x = negative 2.
Noah is not correct because the equivalent expressions mean that there is no solution.
Noah is not correct because he used the distributive property incorrectly.

Answer 1

Noah is correct because the two sides of the equation are equivalent expressions.

In the steps provided:

1. Distributing: \(-3(x + 4) + 2x\) leads to \(-3x - 12 + 2x\), which simplifies to \(-x - 12\).
2. The right side \(2(x - 6) - 3x\) simplifies to \(2x - 12 - 3x\), leading to \(-x - 12\).

So both sides of the equation simplify to \(-x - 12\), which means they are indeed equal for any value of \(x\). This indicates that there are infinitely many solutions because it implies that the equation holds true for all values of \(x\).