Math 136: Solving an Absolute Value Equation

Problem Statement

• Equation to Solve: (9 - |9x| = 8)

Steps to Solve the Equation

Step 1: Isolate the Absolute Value

• Objective: Get the absolute value expression by itself on one side of the equation.
• Initial Equation: (9 - |9x| = 8)
• Process:
  • Subtract 9 from both sides:
    • (9 - 9 = 0)
    • New equation: (-|9x| = -1)
  • Divide both sides by -1 to eliminate the negative:
    • (-|9x| \div -1 = |9x|)
    • (-1 \div -1 = 1)
    • Result: (|9x| = 1)

Step 2: Solve the Absolute Value Equation

• Objective: Split the equation into two separate equations and solve for (x).
• Process:
  1. Positive Case:
     • Remove absolute value: (9x = 1)
     • Solve for (x):
       • Divide by 9: (x = \frac{1}{9})
  2. Negative Case:
     • Make the right side negative: (9x = -1)
     • Solve for (x):
       • Divide by 9: (x = -\frac{1}{9})

Entering Answers in MyMathLab

• Format: Enter results as fractions
• Steps:
  • Enter (\frac{1}{9}) for the positive solution.
  • Use the arrow key to move out of the fraction before adding a comma.
  • Enter (-\frac{1}{9}) for the negative solution.
• Verification: Check the answers in MyMathLab to confirm correctness.

Key Points

• Isolate the absolute value expressions before proceeding.
• Split absolute value equations into two separate linear equations.
• Solve each equation separately to find possible values of (x).