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 infinite solutions. The equation has infinite solutions. 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 no solution.

Answer 1

To solve the equation \( 5(2x + 2) - 4 = 2x + 6 \), we will simplify both sides of the equation and see if it leads to a valid solution.

1. Start with the equation:
\[
5(2x + 2) - 4 = 2x + 6
\]

2. Distribute the 5 on the left side:
\[
10x + 10 - 4 = 2x + 6
\]

3. Combine like terms on the left side:
\[
10x + 6 = 2x + 6
\]

4. Next, isolate the variable \( x \) by subtracting \( 2x \) from both sides:
\[
10x - 2x + 6 = 6
\]
\[
8x + 6 = 6
\]

5. Now, subtract 6 from both sides:
\[
8x = 0
\]

6. Finally, divide by 8:
\[
x = 0
\]

Now we verify the solution:

- Substitute \( x = 0 \) back into the original equation:
\[
5(2(0) + 2) - 4 = 2(0) + 6
\]
Simplifying both sides:
\[
5(0 + 2) - 4 = 0 + 6
\]
\[
10 - 4 = 6
\]
\[
6 = 6
\]

Since both sides are equal, we confirm that \( x = 0 \) is indeed a solution.

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

Hence, the response is: **The equation has one solution: x equals 0.**