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SOLVING ABSOLUTE VALUE INEQUALITIES INVOLVING "LESS THAN"
Jump right to the exercises!
The concepts for this exercise are summarized below.
For a complete discussion, read the text.
MathPlayer is only required for the concept discussion; it is not required for the web exercise.
This section should feel remarkably similar to the previous one.
Instead of solving absolute value equations, this section presents the tools needed
to solve absolute value inequalities involving "less than,"
like these:
|x|<5
|x|≤3
|2-3x
|<7
Each of these inequalities has only a single set of absolute value symbols
which is by itself on the left-hand side of the sentence,
and has a variable inside the absolute value.
The verb is either "<" or
"≤" .
As in the previous section, solving sentences like these is easy,
if you remember the critical fact that
absolute value gives distance from 0 .
Keep this in mind as you read the following theorem:
THEOREM: solving absolute value inequalities involving "less than"
Let
x∈ℝ ,
and let
k≥0 .
Then,
|x|<k
⇔ -k<x<k
|x|≤k
⇔ -k≤x≤k
|
Translating the theorem:
Recall first that normal mathematical conventions dictate
that " |x|<k
"
represents an entire class of sentences,
including |x|<2
,
|x|<5.7 ,
and
|x|<
13 .
The variable k changes from sentence to sentence,
but is constant within a given sentence.
Also,
" -k<x<
k " is a shorthand for
" x>-k
and x<k " ;
i.e., all the numbers between -k and
k .
When you see a sentence of the form
" |x|<k
", here's what you should do:
- Check that k is a positive number.
- The symbol |x| represents the distance between x and 0 .
Thus, you want the numbers x , whose distance from 0 is less than k .
- You can walk from 0 in two directions:
less than k units to the left, or less than
k units to the right.
So, you want all the numbers between -k and
k .
- That is, you want
x>-k
and x<k .
That is, you want
-k<x<
k .
The power of the tool
" |x|<k
⇔ -k<x<k "
goes way beyond solving simple sentences like
" |x|<5 " !
Since x can be any real number,
you should think of x
as merely representing
the stuff inside the absolute value symbols.
Thus, you could think of rewriting the tool as:
" |stuff|<k
⇔ -k<stuff<k " .
See how this idea is used in the following examples:
EXAMPLES:
Solve: |2-3x
|<7
Solution: Write a nice, clean list of equivalent sentences:
| |2-3x
|<7 |
(original sentence) |
| -7<2-
3x<7 |
(use the theorem) |
| -9<-3
x<5 |
(subtract 2 ) |
| 3>x>
-53
|
(divide by -3 ; change direction of inequality symbols) |
| -5
3<x<
3 |
(write in the conventional way) |
Solve:
3|-6x
+7|≤9
Solution: Write a nice, clean list of equivalent sentences:
| 3|-6x
+7|≤9
|
(original sentence) |
| |-6x+
7|≤3
|
(divide by 3 to isolate the absolute value) |
| -3≤-6
x+7≤3
|
(use the theorem) |
| -10≤-
6x≤-4 |
(subtract 7 ) |
| 106
≥x≥
46 |
(divide by -6 ; change the direction of the inequality symbols) |
| 23
≤x≤
53 |
(write in the conventional way) |
Solve:
|5-2x
|<-3
Solution: always false
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.