Question

Solve for x
x
to determine if the equation has one, none, or infinite solutions.
3(2x+2)−4=−2x+10
3
(
2
x
+
2
)
−
4
=
−
2
x
+
10
(1 point)
Responses

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

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

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

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

Answer 1

To solve the equation \(3(2x+2) - 4 = -2x + 10\), let's simplify and solve step-by-step.

1. First, distribute the 3 on the left side:
\[
3(2x) + 3(2) - 4 = -2x + 10
\]
\[
6x + 6 - 4 = -2x + 10
\]
\[
6x + 2 = -2x + 10
\]

2. Next, add \(2x\) to both sides to collect the \(x\) terms:
\[
6x + 2x + 2 = 10
\]
\[
8x + 2 = 10
\]

3. Now, subtract 2 from both sides:
\[
8x = 10 - 2
\]
\[
8x = 8
\]

4. Finally, divide both sides by 8:
\[
x = 1
\]

Now that we have found \(x = 1\), let's determine the nature of the solutions:

Since we have one specific solution, the equation has **one solution: \(x = 1\)**.

So the correct response is:
**The equation has one solution: \(x = 1\).**