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.