Absolute Value Inequalities Tidewater Community College Mr. Joyner

Absolute Value Inequalities First a little review… What does absolute value of a number (or expression) mean?

Absolute Value Inequalities The absolute value of a real number (let’s call it x) is defined as…

Absolute Value Inequalities Writing this a little more symbolically,

Absolute Value Inequalities Wow! That a lot of stuff. What does it all mean?

Absolute Value Inequalities You can think of the absolute value of a real number as the answer to the question … How far does this real number lie from zero (the origin) on the real number line?

Absolute Value Inequalities Or more simply… What is the distance between zero and this number?

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 have two inequality senses (directions) to deal with: We only have one sense (direction) to deal with for an equation ( = ), but … 1. greater than ( > ) 2. 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 are all the values of x that lie between -a AND a. Case 1. Remember, we need the “distance” of x from zero to be less than the value a.

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

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

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

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

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

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: Check: 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.

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 Case 2 Example: Check: 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. or Go to Practice Problems