Question

Solve for x to determine if the equation has one, none, or infinite solutions.
5(2x+2)-7x=3x+6
The equation has infinite solutions.
The equation has one solution: x=6
The equation has one solution:x=10.
The equation has no solution.

Answer 1

To solve the equation \( 5(2x + 2) - 7x = 3x + 6 \), we will first simplify the left-hand side.

1. Distribute \( 5 \):
\[
5(2x + 2) = 10x + 10
\]
So, the equation becomes:
\[
10x + 10 - 7x = 3x + 6
\]

2. Combine like terms on the left-hand side:
\[
(10x - 7x) + 10 = 3x + 6
\]
which simplifies to:
\[
3x + 10 = 3x + 6
\]

3. Now, we can subtract \( 3x \) from both sides:
\[
10 = 6
\]

4. This statement is false.

Since we arrived at a false statement, it means that there are **no solutions** to the given equation. Therefore, the correct conclusion is:

**The equation has no solution.**