Question

Solve for x
𝑥
to determine if the equation has one, none, or infinite solutions.
5(2x+2)−4=2x+6
5
(
2
𝑥
+
2
)
−
4
=
2
𝑥
+
6
(1 point)
Responses

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

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

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

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

Answer 1

To solve the equation \( 5(2x + 2) - 4 = 2x + 6 \), we start by simplifying both sides step-by-step.

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

Now, the equation looks like this:
\[
10x + 6 = 2x + 6
\]

2. Next, we want to isolate the \( x \) terms. Subtract \( 2x \) from both sides:
\[
10x - 2x + 6 = 6
\]
which simplifies to:
\[
8x + 6 = 6
\]

3. Now, subtract 6 from both sides:
\[
8x + 6 - 6 = 6 - 6
\]
This results in:
\[
8x = 0
\]

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

Now, we can check the original equation to determine if it has a unique solution, infinite solutions, or no solutions:
- Substitute \( x = 0 \) back into the original equation:
\[
5(2(0) + 2) - 4 = 2(0) + 6
\]
Simplifying both sides yields:
\[
5(2) - 4 = 6 \implies 10 - 4 = 6 \implies 6 = 6
\]

Since both sides of the equation are equal when \( x = 0 \), we have verified that there is a single solution.

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