Podcast
Questions and Answers
What is the absolute value of a number?
What is the absolute value of a number?
What is the property of absolute value that states |a| ≥ 0 for all real numbers a?
What is the property of absolute value that states |a| ≥ 0 for all real numbers a?
What is the equivalent form of the inequality |x| < a?
What is the equivalent form of the inequality |x| < a?
What is the equivalent form of the inequality |x| > a?
What is the equivalent form of the inequality |x| > a?
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?
Why is it helpful to graph the related function on a number line when solving inequalities involving absolute values?
Signup and view all the answers
What should you be careful about when solving inequalities involving absolute values?
What should you be careful about when solving inequalities involving absolute values?
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| = 5and|-3| = 3.
Properties of Absolute Value
-
|a| ≥ 0for all real numbersa. -
|a| = 0if and only ifa = 0. -
|ab| = |a| |b|for all real numbersaandb. -
|a + b| ≤ |a| + |b|for all real numbersaandb.
Absolute Value in Inequalities
- The absolute value of an expression can be used to solve inequalities involving absolute values.
- The basic rule is:
|x| < ais equivalent to-a < x < a. - Similarly,
|x| > ais equivalent tox < -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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.
