Transcript Absolute Value Inequalities - Tidewater Community College

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…
x

 x , for x greater than or equal to


 x , for x less than zero
zero
Absolute Value Inequalities
Writing this a little
more symbolically,
x
 x , x  0

 x , x  0
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…
6  6
6  6
0 0
Absolute Value Inequalities
When solving an absolute value
equation, there are always two
cases to consider.
In solving x  8
there are two values of x
that are solutions.
Absolute Value Inequalities
x 8  x 8
or x  8
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:
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
Case 1.
The solutions of
x a
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
Case 1.
The solutions of
x a
Where do we find such values
on the real number line?
Absolute Value Inequalities
Case 1.
Symbolically, we write
the solutions of x  a
as  a
xa
Absolute Value Inequalities
Case 2.
The solutions of
x a
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
Case 2.
The solutions of
x a
Where do we find such values
on the real number line?
Absolute Value Inequalities
Case 2.
Symbolically, we write
the solutions of x  a
as
x  a
OR
xa
Absolute Value Inequalities
Case 1 Example:
x  3  5
x  2
x 3  5
and
x 3  5
x8
2  x  8
Absolute Value Inequalities
Case 1 Alternate
method:
x 3  5
The two statements: x  3  5, and, x  3  5
can be written using a shortened version which I
call a triple inequality
5  x 3  5
This shortened version can
only be used for absolute
value less than problems.
It is not appropriate for
the greater than problems.
5 3  x 33  5 3
 2  x  8 This is the preferred method.
Absolute Value Inequalities
Case 1 Example: x  3  5
 2  x  8
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.
x 5  7
x  5  7, Switch_& _ Negate
or
x  5  7, remove _ abs.val_ symbols
Absolute Value Inequalities
Case 2 Example: 2 x  1  9
2x  1   9
2x   10
x  5
or
2x  1  9
2x  8
x4
x   5 OR x  4
Absolute Value Inequalities
Case 2 Example: 2 x  1  9  x   5 or x  4
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.
Go to Practice Problems