Solving Absolute Value Equations

Steps to Solve

1. Isolate the Absolute Value
   • Move absolute value term to one side of the equation.
2. Set Up Two Equations
   • Drop the absolute value and solve for both the positive and negative scenarios.
3. Solve and Check Solutions
   • Solve the two resulting equations.
   • Verify solutions by substituting back into the original equation.

Example 1

• Equation: (-2 \times |x + 4| = -10)

  Steps:

  • Isolate: Divide both sides by (-2) to get (|x + 4| = 5).
  • Set Up Equations:
    • (x + 4 = 5)
    • (x + 4 = -5)
  • Solve Equations:
    • For (x + 4 = 5): Subtract 4 from both sides -> (x = 1)
    • For (x + 4 = -5): Subtract 4 from both sides -> (x = -9)
  • Check Solutions:
    • (x = 1): (-2 \times 5 = -10) (True)
    • (x = -9): (-2 \times 5 = -10) (True)

Example 2

• Equation: (|3x - 1| - 9 = 4)

  Steps:

  • Isolate: Add 9 to both sides to get (|3x - 1| = 13).
  • Set Up Equations:
    • (3x - 1 = 13)
    • (3x - 1 = -13)
  • Solve Equations:
    • For (3x - 1 = 13):
      • Add 1 to both sides -> (3x = 14)
      • Divide by 3 -> (x = 14/3)
    • For (3x - 1 = -13):
      • Add 1 to both sides -> (3x = -12)
      • Divide by 3 -> (x = -4)
  • Check Solutions: Verify by substitution.

Key Concepts

• Absolute Value Principle:
  • If (|A| = B), then (A = B) or (A = -B).
• Verification: Always check solutions by substituting back into the original equation to ensure validity.

These steps can be applied to solve any absolute value equations systematically by isolating the absolute value, creating equivalent equations, solving them, and verifying the potential solutions.