Solve the absolute value equation |12−3x|=|9+4x| .(1 point) Responses no solution no solution x=4 x equals 4 x=−21 and x=37 x equals negative 21 and x equals Start Fraction 3 over 7 End Fraction x=37 and x=3

Answer 1

To solve the absolute value equation \( |12 - 3x| = |9 + 4x| \), we need to consider the different cases based on the properties of absolute values.

Case 1: \( 12 - 3x = 9 + 4x \)

1. Solve for \( x \): \[ 12 - 3x = 9 + 4x \] \[ 12 - 9 = 4x + 3x \] \[ 3 = 7x \] \[ x = \frac{3}{7} \]

Case 2: \( 12 - 3x = -(9 + 4x) \)

1. Solve for \( x \): \[ 12 - 3x = -9 - 4x \] \[ 12 + 9 = -4x + 3x \] \[ 21 = -x \] \[ x = -21 \]

Case 3: \( -(12 - 3x) = 9 + 4x \)

1. Solve for \( x \): \[ -12 + 3x = 9 + 4x \] \[ -12 - 9 = 4x - 3x \] \[ -21 = x \] (This is the same as Case 2.)

Case 4: \( -(12 - 3x) = -(9 + 4x) \)

1. Solve for \( x \): \[ -12 + 3x = -9 - 4x \] \[ -12 + 9 = -4x - 3x \] \[ -3 = -7x \] \[ x = \frac{3}{7} \] (This is the same as Case 1.)

Summary of Solutions

From these cases, we have the solutions:

• \( x = \frac{3}{7} \)
• \( x = -21 \)

Final Answer

The solutions to the equation \( |12 - 3x| = |9 + 4x| \) are \( x = \frac{3}{7} \) and \( x = -21 \).

If we refer to the response options, the correct one is:

x = −21 and x = 3/7