Absolute Value in Inequality

Absolute Value in Inequality

Definition

• The absolute value of a number is its distance from zero on the number line.
• It is denoted by the symbol | |.
• For example, |5| = 5 and |-3| = 3.

Properties of Absolute Value

• |a| ≥ 0 for all real numbers a.
• |a| = 0 if and only if a = 0.
• |ab| = |a| |b| for all real numbers a and b.
• |a + b| ≤ |a| + |b| for all real numbers a and b.

Absolute Value in Inequalities

• The absolute value of an expression can be used to solve inequalities involving absolute values.
• The basic rule is: |x| < a is equivalent to -a < x < a.
• Similarly, |x| > a is equivalent to x < -a or x > a.
• When solving inequalities involving absolute values, it is often helpful to break them down into two separate inequalities, one where the expression inside the absolute value is positive and one where it is negative.

Examples

• Solve |x - 2| < 3:
  • -3 < x - 2 < 3
  • -1 < x < 5
• Solve |x + 1| > 2:
  • x + 1 > 2 or x + 1 < -2
  • x > 1 or x < -3

Tips and Tricks

• When solving inequalities involving absolute values, it is often helpful to graph the related function on a number line to visualize the solution.
• Be careful when solving inequalities involving absolute values, as the direction of the inequality can change depending on the sign of the expression inside the absolute value.