Solving Absolute Value Equations and Inequalities

Solving Absolute Value Equations

• Definition: The absolute value of a number is its distance from zero on the number line, always non-negative.
• Standard Form: |x| = a, where a ≥ 0.
• General Solution:
  • If |x| = a, then the solutions are:
    • x = a
    • x = -a
• Example:
  • For |x| = 3:
    • Solutions: x = 3 or x = -3
• Steps to Solve:
  1. Isolate the absolute value expression.
  2. Set up two separate equations (one for the positive case and one for the negative).
  3. Solve each equation individually.
  4. Check both solutions in the original equation.

Inequalities Involving Absolute Value

• Form: |x| < a or |x| > a, where a ≥ 0.
• For |x| < a (Within the bounds):
  • The solution is: -a < x < a
• For |x| > a (Outside the bounds):
  • The solution is: x < -a or x > a
• Example:
  • |x| < 4:
    • Solutions: -4 < x < 4
  • |x| > 2:
    • Solutions: x < -2 or x > 2
• Steps to Solve Absolute Value Inequalities:
  1. Isolate the absolute value expression.
  2. Set up the appropriate compound inequality based on the inequality sign.
  3. Solve the resulting inequalities.
  4. Graph the solution on a number line to visualize.

Key Points to Remember

• When handling equations, always confirm solutions fit the original.
• For inequalities, consider both scenarios (less than and greater than) clearly and check for openings or closures on intervals.
• Graphical representation can aid in understanding the solution sets for inequalities.

Absolute Value Equations

• The absolute value of a number represents its distance from zero on the number line, always a non-negative value.
• The standard form for an absolute value equation is |x| = a, where 'a' is greater than or equal to zero.
• To solve an absolute value equation, we set up two separate equations: one for the positive value ('x' = 'a') and one for the negative value ('x' = - 'a').
• Always check if the solutions obtained satisfy the original equation.

Absolute Value Inequalities

• Absolute value inequalities follow the form |x| < a or |x| > a, with 'a' being greater than or equal to zero.
• When |x| < a, the solution lies within the bounds, represented as -a < x < a.
• When |x| > a, the solution falls outside the bounds, expressed as x < -a or x > a.
• To solve these inequalities, isolate the absolute value expression and set up the appropriate compound inequality (for less than or greater than).
• Visualizing the solution on a number line is helpful for understanding the intervals.

Key Points

• Always verify that the solutions found for equations satisfy the original absolute value equation.
• When solving inequalities, carefully consider both scenarios: less than and greater than.
• For inequalities, it's important to note whether intervals are open (exclusive) or closed (inclusive), denoted by parenthesis or brackets respectively.
• Visual representation through graphs can aid in understanding the solution sets for absolute value inequalities.

This quiz covers the concepts of absolute value equations and inequalities, including their definitions, standard forms, general solutions, and steps to solve. It provides examples for clarity and guides you through solving both equations and inequalities involving absolute values.