Question

Solve for x
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.**