Solve for x to determine the equation has one, none, or infinite solutions.

Question

3(2x + 2) - 4 = -2x + 10
A. The equation has one solution: x = 0.
B. The equation has no solution.
C. The equation has one solution: x = 1.
D. The equation has infinite solutions.

Answer 1

To solve the equation \( 3(2x + 2) - 4 = -2x + 10 \), we will first simplify each side of the equation step by step.

1. Distribute on the left side:

   \[ 3(2x + 2) = 6x + 6 \]

   So, the left side becomes:

   \[ 6x + 6 - 4 = 6x + 2 \]

   Therefore, the equation simplifies to:

   \[ 6x + 2 = -2x + 10 \]

2. Next, we will add \( 2x \) to both sides to get all \( x \) terms on one side:

   \[ 6x + 2 + 2x = 10 \]

   Which simplifies to:

   \[ 8x + 2 = 10 \]

3. Now, subtract \( 2 \) from both sides:

   \[ 8x = 8 \]

4. Finally, divide by \( 8 \):

   \[ x = 1 \]

Now, let's determine the nature of the solution. Since we found that \( x = 1 \) is the only solution, we conclude that:

The equation has one solution: \( x = 1 \).

Therefore, the correct answer is:

C. The equation has one solution: x = 1.