absolute value inequalities tidewater community college mr. joyner
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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…
, for x greater than or equal to zero, for x less than zero
xx
x
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
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.
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
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.
Absolute Value Inequalities
ax The solutions of
Where do we find such values on the real number line?
Case 1.
Absolute Value Inequalities
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.
Absolute Value Inequalities
ax The solutions of
Where do we find such values on the real number line?
Case 2.
Absolute Value Inequalities
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.
Absolute Value Inequalities
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
_._,
__&,
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.
2 1 9x 5x 4x or
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