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1.6 – Absolute Value Equations and Inequalities

Learn to write, solve absolute value equations and inequalities. Understand extraneous solutions, graph solutions, and compound inequalities. Check for extraneous solutions and describe tolerance using absolute value. Includes practice problems.

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1.6 – Absolute Value Equations and Inequalities

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  1. 1.6 – Absolute Value Equations and Inequalities Students will be able to: Write and solve equations and inequalities involving absolute value. Lesson Vocabulary Absolute value Extraneous solution

  2. 1.6 – Absolute Value Equations and Inequalities An absolute value quantity is nonnegative. Since opposites have the same absolute value, an absolute value equation CAN have two solutions.

  3. 1.6 – Absolute Value Equations and Inequalities An absolute value equation has a variable within the absolute value sign. For example, |x| = 5. Here, the value of x can be 5 or -5 since |5| = 5 and |-5| = 5.

  4. 1.6 – Absolute Value Equations and Inequalities Problem 1: What is the solution of |2x – 1| = 5? Graph the solution.

  5. 1.6 – Absolute Value Equations and Inequalities Problem 1b: What is the solution of |3x +2| = 4? Graph the solution.

  6. 1.6 – Absolute Value Equations and Inequalities Problem 1c: What is the solution of |5x +2| = -7? Graph the solution.

  7. 1.6 – Absolute Value Equations and Inequalities Problem 2: What is the solution of 3|x +2| - 1 = 8? Graph the solution.

  8. 1.6 – Absolute Value Equations and Inequalities Problem 2b: What is the solution of 2|x +9| +3 = 7? Graph the solution.

  9. 1.6 – Absolute Value Equations and Inequalities An extraneous solution is a solution derived from an original equations that is NOT a solution of the original equation.

  10. 1.6 – Absolute Value Equations and Inequalities Problem 3: What is the solution of |3x +2| = 4x + 5? Check for extraneous solutions!

  11. 1.6 – Absolute Value Equations and Inequalities Problem 3b: What is the solution of |5x - 2| = 7x + 14? Check for extraneous solutions!

  12. 1.6 – Absolute Value Equations and Inequalities The solutions of the absolute value inequality |x| < 5 include values greater than -5 and less than 5. This is a compound inequality x>-5 and x<5, which you can write as -5 < x < 5. So |x| < 5 means x is BETWEEN -5 and 5.

  13. 1.6 – Absolute Value Equations and Inequalities Problem 4: What is the solution of |2x - 1| < 5? Graph the solution.

  14. 1.6 – Absolute Value Equations and Inequalities Is this an “and” problem or an “or” problem? Problem 4b: What is the solution of |3x - 4| < 8? Graph the solution.

  15. 1.6 – Absolute Value Equations and Inequalities Is this an “and” problem or an “or” problem? Problem 5: What is the solution of |2x +4| > 6? Graph the solution.

  16. 1.6 – Absolute Value Equations and Inequalities Is this an “and” problem or an “or” problem? Problem 5b: What is the solution of |5x +10| > 15? Graph the solution.

  17. 1.6 – Absolute Value Equations and Inequalities Problem 5c: Without solving |x – 3| > 2, describe the graph of its solution.

  18. 1.6 – Absolute Value Equations and Inequalities A manufactured item’s actual measurements and its target measurements can differ by a certain amount, called tolerance. Tolerance is one half the difference of the maximum and minimum acceptable values. You can use absolute value inequalities to describe tolerance.

  19. 1.6 – Absolute Value Equations and Inequalities Problem 6: In car racing, a car must meet specific dimensions to enter the race. Officials use a template to ensure these specifications are met. What absolute value inequality describes heights of the model race car shown within the indicated tolerance?

  20. 1.6 – Absolute Value Equations and Inequalities Exit Ticket: Explain what it means for a solution of an equation to be extraneous. When is the absolute value of a number equal to the number itself? Give an example of a compound inequality that has no solution.

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