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

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.

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.

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

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

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

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

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

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.

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

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

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.

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

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

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

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

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

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.

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?

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.