Absolute Value Inequalities Tidewater Community College Mr. Joyner

Absolute Value Inequalities

Tidewater Community CollegeMr. Joyner


First a little review…

What does absolute value of a number (or expression) mean?


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

, for x greater than or equal to zero, for x less than zero

xx

x


Writing this a little more symbolically,

, 0, 0

x xx x

x


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


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?


Or more simply…

What is the distance between zero and this number?


Examples… 6 6

6 6

0 0


When solving an absolute value equation, there are always two cases to consider.

8x In solving

there are two values of x that are solutions.


8x

because the absolute value of both numbers is 8.

8 or 8x x


OK, now on to 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 ( < )


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


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


ax 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.


ax The solutions of

Where do we find such values on the real number line?

Case 1.


ax Symbolically, we write the solutions of

Case 1.

a x a as


ax 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.


ax The solutions of

Where do we find such values on the real number line?

Case 2.

Symbolically, we write the solutions of


ax

Case 2.

x a as x aOR


Case 1 Example:

3 5x

3 5

2

x

x

and

8x

53x

8x2


Case 1 Alternate method:

3 5x

The two statements: 53xand53x ,,

can be written using a shortened version which I call a triple inequality

8x2

3533x35

53x5

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.


Case 1 Example:

3 5x

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.

8x2

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.

symbolsvalabsremove75x

or

NegateSwitch75x

75x

_._,

__&,


Case 2 Example:

2 1 9x

2 1 9

2 10

5

x

x

x

5x

2 1 9

2 8

4

x

x

x

4x OR

or


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.

2 1 9x 5x 4x or

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Absolute Value Inequalities Tidewater Community College Mr. Joyner