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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.

1 answer

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.