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Solve the absolute value equation |12−3x|=|9+4x| .(1 point)

Responses

x=4
x equals 4

x=−21 and x=3/7
x equals negative 21 and x equals Start Fraction 3 over 7 End Fraction

x=3/7 and x=3
x equals Start Fraction 3 over 7 End Fraction and x equals 3

no solution

1 answer

To solve the equation \( |12 - 3x| = |9 + 4x| \), we need to consider the different cases that arise from the absolute values.

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

  1. Combine like terms: \[ 12 - 9 = 4x + 3x \] \[ 3 = 7x \quad \Rightarrow \quad x = \frac{3}{7} \]

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

  1. Expand and combine like terms: \[ 12 - 3x = -9 - 4x \] \[ 12 + 9 = -4x + 3x \] \[ 21 = -x \quad \Rightarrow \quad x = -21 \]

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

  1. Expand and combine like terms: \[ -12 + 3x = 9 + 4x \] \[ -12 - 9 = 4x - 3x \] \[ -21 = x \]

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

  1. This leads to: \[ -12 + 3x = -9 - 4x \] \[ -12 + 9 = -4x - 3x \] \[ -3 = -7x \quad \Rightarrow \quad x = \frac{3}{7} \]

Summary of solutions:

The solutions from our cases are:

  • From Case 1: \( x = \frac{3}{7} \)
  • From Case 2: \( x = -21 \)
  • From Case 3: \( x = -21 \) (duplicate)
  • From Case 4: \( x = \frac{3}{7} \) (duplicate)

The unique solutions to the equation \( |12 - 3x| = |9 + 4x| \) are:

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

Final Answer:

The correct response is: x equals negative 21 and x equals Start Fraction 3 over 7 End Fraction.