4. Select the equation that has infinitely many solutions.(1 point)Responses4x+1=4(x+1)4 x plus 1 is equal to 4 times open paren x plus 1 close paren3−3x=−3(x−1)3 minus 3 x is equal to negative 3 times open paren x minus 1 close paren3x=2x+13 x is equal to 2 x plus 13x+5=3x−5

Answer 1

To determine which equation has infinitely many solutions, we need to identify the equations that are equivalent or simplify to a true statement regardless of the value of \( x \).

Let's analyze each option:

1. \( 4x + 1 = 4(x + 1) \):

   • Simplifying the right side: \[ 4(x + 1) = 4x + 4 \]
   • Thus, the equation becomes: \[ 4x + 1 = 4x + 4 \]
   • Subtracting \( 4x \) from both sides: \[ 1 = 4 \] (false)

2. \( 3 - 3x = -3(x - 1) \):

   • Simplifying the right side: \[ -3(x - 1) = -3x + 3 \]
   • The equation becomes: \[ 3 - 3x = -3x + 3 \]
   • Adding \( 3x \) to both sides: \[ 3 = 3 \] (true, so infinitely many solutions)

3. \( 3x = 2x + 1 \):

   • Subtracting \( 2x \) from both sides: \[ x = 1 \] (one solution)

4. \( 3x + 5 = 3x - 5 \):

   • Subtracting \( 3x \) from both sides: \[ 5 = -5 \] (false)

The only equation that simplifies to a true statement (indicating infinitely many solutions) is:

\( 3 - 3x = -3(x - 1) \).