Recording-2025-01-09T18:40:54.501Z

Absolute Value Equations

Types of Solutions

• Absolute value equations can have:

  • Two Solutions

  • One Solution

  • No Solutions

Two Solution Equations

• Example:

  • Equation: (-|x| = -5)

  • Isolate Absolute Value:

    • Divide both sides by -1:

    • (|x| = 5)

  • Possible Solutions:

    • (x = -5)

    • (x = 5)

One Solution Equations

• Example:

  • Equation: (|x| = 0)

  • Reasoning:

    • Distance of 0 on the number line

    • Only one solution:

    • (x = 0)

No Solution Equations

• Example:

  • Equation: (|x| = -5)

  • Reasoning:

    • Absolute value represents distance from 0

    • Cannot have a negative distance

    • Thus, this equation has no solution

Distinction Between Forms

• Alternate Form:

  • When expressed as (-|x| = -5):

    • Isolate absolute value by converting to positive:

    • Divide both sides by -1:

    • This leads to (|x| = 5) (which then has two solutions)

• Important Note:

  • The negative sign in front of the absolute value affects the equality differently than if the absolute value is directly set to a negative number.