Abs Value Inequalities 2

7/27/2019 Abs Value Inequalities 2

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Absolute Value Inequalities

Tidewater Community College

Mr. Joyner


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First a little review

What does absolute value of anumber (or expression) mean?


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The absolute value of a realnumber (lets call it x) is

defined as, for x greater than or equal to zero

, for x less than zero

x

x

x


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Writing this a littlemore symbolically,

, 0

, 0

x x

x x

x


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Wow! That a lot ofstuff. What does itall mean?


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You can think of the absolutevalue of a real number as the

answer to the question How far does this real number

lie from zero (the origin) onthe real number line?


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Or more simply

What is the distance between zeroand this number?


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Examples 6 6

6 6

0 0


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When solving an absolute valueequation, there are always two

cases to consider.

8x In solving

there are two values of xthat are solutions.


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8x

because the absolute value ofboth numbers is 8.

8 or 8x x


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OK, now on to absolute valueinequalities.


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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 ( < )


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In solving an absolute valueinequality, we have to treat the

two inequality senses separately.


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For a real number variable orexpression (lets call it x) and a

non-negative, real number (letscall it a)


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ax The solutions of

are all the values of x that liebetween -a AND a.

Case 1.

Remember, we need thedistance of x from zero to beless than the value a.


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ax The solutions of

Where do we find such valueson the real number line?

Case 1.


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ax

Symbolically, we write

the solutions of

Case 1.

a x a as


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ax The solutions of

are all the values of x that areless than a OR greater than a.

Case 2.

Remember, we need thedistance of x from zero tobe greater than the value a.


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ax The solutions of

Where do we find such valueson the real number line?

Case 2.


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Symbolically, we write

the solutions of


ax

Case 2.

x a as x aOR


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Case 1 Example: 3 5x

3 5

2

x

x

and

8x53x

8x2


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Case 1 Alternatemethod:

3 5x

The two statements: 53xand53x ,,

can be written using a shortened version which Icall a triple inequality

8x23533x35

53x5

This shortened version canonly be used for absolutevalue less than problems.It is not appropriate for

the greater than problems.

This is the preferredmethod.


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Case 1 Example: 3 5x

Check: Choose a value of x in the solutioninterval, say x = 1, and test it to makesure that the resulting statement is true.Choose a value of x NOT in the solution

interval, say x = 9, and test it to makesure that the resulting statement is false.

8x2


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Things to remember:Absolute Value problems that are less than have anand solution and can be written as a triple inequality.

Absolute Value problems that are greater than have anor solution and must be written as two separate

inequalities.Theway to remember how to write the two inequalitiesis: for one statement switch the order symbol andnegate the number, for the other just remove the abs

value symbols.

symbolsvalabsremove75x

or

NegateSwitch75x

75x

_._,

__&,


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Case 2 Example: 2 1 9x

2 1 92 10

5

x

x

x

5x

2 1 92 8

4

x

x

x

4x OR

or


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Case 2 Example:

Check: Choose a value of x in the solutionintervals, say x = -8, and test it to makesure that the resulting statement is true.Choose a value of x NOT in the solution

interval, say x = 0, and test it to makesure that the resulting statement is false.

2 1 9x 5x 4x or

Go to Practice Problems
http://localhost/var/www/apps/conversion/tmp/scratch_3/Abs%20Value%20Inequalities%20Practice%20Problems.ppthttp://localhost/var/www/apps/conversion/tmp/scratch_3/Abs%20Value%20Inequalities%20Practice%20Problems.ppt

Abs Value Inequalities 2

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