Absolute Value Inequalities Tidewater Community College Mr. Joyner

Absolute Value Inequalities Examples…

Absolute Value Inequalities When solving an absolute value equation, there are always two cases to consider. In solving there are two values of x that are solutions.

Absolute Value Inequalities because the absolute value of both numbers is 8.

Absolute Value Inequalities OK, now on to absolute value inequalities.

Absolute Value Inequalities We only have one sense (direction) to deal with for an equation ( = ) , but … we have two inequality senses (directions) to deal with: greater than ( > ) less than ( < )

Absolute Value Inequalities In solving an absolute value inequality, we have to treat the two inequality senses separately.

Absolute Value Inequalities For a real number variable or expression (let’s call it x) and a non-negative, real number (let’s call it a)…

Absolute Value Inequalities The solutions of Case 1. are all the values of x that lie between -a AND a. Remember, we need the “distance” of x from zero to be less than the value a.

Absolute Value Inequalities The solutions of Case 1. Where do we find such values on the real number line?

Absolute Value Inequalities Case 1. Symbolically, we write the solutions of as

Absolute Value Inequalities The solutions of Case 2. are all the values of x that are less than –a OR greater than a. Remember, we need the “distance” of x from zero to be greater than the value a.

Absolute Value Inequalities The solutions of Case 2. Where do we find such values on the real number line?

Absolute Value Inequalities Case 2. Symbolically, we write the solutions of as OR

Absolute Value Inequalities Case 1 Example: and

Absolute Value Inequalities Case 1 Alternate method: The two statements: can be written using a shortened version which I call a triple inequality This shortened version can only be used for absolute value less than problems. It is not appropriate for the greater than problems. This is the preferred method.

Absolute Value Inequalities Case 1 Example: Choose a value of x in the solution interval, say x = 1, and test it to make sure that the resulting statement is true. Choose a value of x NOT in the solution interval, say x = 9, and test it to make sure that the resulting statement is false. Check:

Things to remember: Absolute Value problems that are “less than” have an “and” solution and can be written as a triple inequality. Absolute Value problems that are “greater than” have an “or” solution and must be written as two separate inequalities. The way to remember how to write the two inequalities is: for one statement switch the order symbol and negate the number, for the other just remove the abs value symbols.

Absolute Value Inequalities Case 2 Example: or OR

Absolute Value Inequalities or Case 2 Example: Choose a value of x in the solution intervals, say x = -8, and test it to make sure that the resulting statement is true. Choose a value of x NOT in the solution interval, say x = 0, and test it to make sure that the resulting statement is false. Check: