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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 one solution: x=1 . The equation has one solution: x equals 1 . The equation has no solution. The equation has no solution. The equation has one solution: x=0 . The equation has one solution: x equals 0 . The equation has infinite solutions.

1 answer

Let's solve the equation step by step:

The equation we have is:

\[ 5(2x + 2) - 4 = 2x + 6 \]

  1. Distribute \(5\) on the left side:

\[ 10x + 10 - 4 = 2x + 6 \]

  1. Simplify the left side:

\[ 10x + 6 = 2x + 6 \]

  1. Now, subtract \(2x\) from both sides:

\[ 10x - 2x + 6 = 6 \]

This simplifies to:

\[ 8x + 6 = 6 \]

  1. Next, subtract \(6\) from both sides:

\[ 8x = 0 \]

  1. Now, divide both sides by \(8\):

\[ x = 0 \]

Since we have found a unique solution \(x = 0\), we check the options provided:

  • The equation has one solution: x=1.
  • The equation has one solution: x equals 1.
  • The equation has no solution.
  • The equation has no solution.
  • The equation has one solution: x=0.
  • The equation has one solution: x equals 0.
  • The equation has infinite solutions.

The correct response is:

The equation has one solution: x equals 0.