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

Question

3(2x+2)−4=−2x+10 (1 point)
Responses

The equation has no solution.
The equation has no solution.

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

The equation has one solution: x=0.
The equation has one solution: x equals 0 .

The equation has infinite solutions.

Answer 1

To solve the equation \( 3(2x + 2) - 4 = -2x + 10 \), let's first simplify each side.

1. Start with the left-hand side: \[ 3(2x + 2) - 4 = 6x + 6 - 4 = 6x + 2 \]

2. Now write the equation: \[ 6x + 2 = -2x + 10 \]

3. Next, let's add \( 2x \) to both sides: \[ 6x + 2 + 2x = 10 \] \[ 8x + 2 = 10 \]

4. Now, subtract \( 2 \) from both sides: \[ 8x = 8 \]

5. Finally, divide by \( 8 \): \[ x = 1 \]

The solution to the equation is \( x = 1 \).

Now, we check the responses:

• The equation has no solution. (Incorrect)
• The equation has one solution: \( x = 1 \). (Correct)
• The equation has one solution: \( x = 0 \). (Incorrect)
• The equation has infinite solutions. (Incorrect)

Thus, the correct response is: The equation has one solution: x equals 1.