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

The equation has infinite solutions.
The equation has infinite solutions.

The equation has one solution: x=1
.

Answer 1

To solve the equation \( 5(2x + 2) - 4 = 2x + 6 \), we will first simplify both sides.

1. Distribute the \( 5 \) on the left side: \[ 5(2x) + 5(2) - 4 = 10x + 10 - 4 = 10x + 6 \] So we rewrite the equation: \[ 10x + 6 = 2x + 6 \]

2. Next, we eliminate \( 2x \) from both sides: \[ 10x + 6 - 2x = 2x + 6 - 2x \] This simplifies to: \[ 8x + 6 = 6 \]

3. Now, subtract \( 6 \) from both sides: \[ 8x + 6 - 6 = 6 - 6 \] Thus: \[ 8x = 0 \]

4. Finally, divide both sides by \( 8 \): \[ x = 0 \]

Now let's check the possibilities for the number of solutions:

• If we plug \( x = 0 \) back into the original equation: \[ 5(2(0) + 2) - 4 = 2(0) + 6 \] This simplifies to: \[ 5(0 + 2) - 4 = 0 + 6 \] Which gives: \[ 10 - 4 = 6 \] This results in \( 6 = 6 \), which is true.

Since we have found a solution (and it was the only one), we conclude:

The equation has one solution: \( x = 0 \).