Solve for x

Question

to determine if the equation has one, none, or infinite solutions.
5(2x+2)−4=2x+6
(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 \(5(2x + 2) - 4 = 2x + 6\), we can start by expanding and simplifying both sides.

1. Expand the left side: \[ 5(2x + 2) - 4 = 10x + 10 - 4 = 10x + 6 \]

2. Rewrite the equation: \[ 10x + 6 = 2x + 6 \]

3. Subtract \(2x\) from both sides: \[ 10x - 2x + 6 = 6 \] This simplifies to: \[ 8x + 6 = 6 \]

4. Subtract 6 from both sides: \[ 8x = 0 \]

5. Divide by 8: \[ x = 0 \]

Now we check the solution:

Substituting \(x = 0\) back into the original equation:

Left side: \[ 5(2(0) + 2) - 4 = 5(0 + 2) - 4 = 5 \cdot 2 - 4 = 10 - 4 = 6 \]

Right side: \[ 2(0) + 6 = 0 + 6 = 6 \]

Both sides equal 6, confirming that our solution \(x = 0\) is correct.

Since we found one unique solution, the answer is:

The equation has one solution: x=0.