To solve the equation \( |12 - 3x| = |9 + 4x| \), we need to consider the different cases for absolute values.
We can break this down into four cases based on the expressions inside the absolute values.
Case 1:
\( 12 - 3x = 9 + 4x \)
Solving this: \[ 12 - 3x = 9 + 4x \] \[ 12 - 9 = 4x + 3x \] \[ 3 = 7x \] \[ x = \frac{3}{7} \]
Case 2:
\( 12 - 3x = -(9 + 4x) \)
Solving this: \[ 12 - 3x = -9 - 4x \] \[ 12 + 9 = -4x + 3x \] \[ 21 = -x \] \[ x = -21 \]
Case 3:
\( -(12 - 3x) = 9 + 4x \)
Solving this: \[ -12 + 3x = 9 + 4x \] \[ -12 - 9 = 4x - 3x \] \[ -21 = x \] This is a repeated solution, so we will only take \( x = -21 \).
Case 4:
\( -(12 - 3x) = -(9 + 4x) \)
Solving this: \[ -12 + 3x = -9 - 4x \] \[ -12 + 9 = -4x - 3x \] \[ -3 = -7x \] \[ x = \frac{3}{7} \] This is a repeated solution, so we will only take \( x = \frac{3}{7} \).
Summary of Solutions
The solutions we found are:
- \( x = -21 \)
- \( x = \frac{3}{7} \)
Thus the answers to the equations \( |12 - 3x| = |9 + 4x| \) are:
- \( x = -21 \)
- \( x = \frac{3}{7} \)
The correct response is: x equals negative 21 and x equals Start Fraction 3 over 7 End Fraction.