Absolute Value in Inequality
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Questions and Answers

What is the absolute value of a number?

  • Its square root
  • Its reciprocal
  • Its distance from zero on the number line (correct)
  • Its cube root

    • What is the property of absolute value that states |a| ≥ 0 for all real numbers a?

  • Property of non-negativity (correct)
  • Property of symmetry
  • Property of multiplicativity
  • Property of additivity

    • What is the equivalent form of the inequality |x| < a?

  • a > x > -a
  • a < x < -a
  • -a < x < a (correct)
  • -a < x > a

    • What is the equivalent form of the inequality |x| > a?

    <p>x &gt; a or x &lt; -a</p> Signup and view all the answers

    Why is it helpful to graph the related function on a number line when solving inequalities involving absolute values?

    <p>To visualize the solution</p> Signup and view all the answers

    What should you be careful about when solving inequalities involving absolute values?

    <p>The sign of the expression inside the absolute value</p> Signup and view all the answers

    Study Notes

    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.

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    Description

    Learn about the definition and properties of absolute value, and how to solve inequalities involving absolute values. Practice solving examples and learn tips and tricks to master absolute value inequalities.

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