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SOLVING ABSOLUTE VALUE EQUATIONS
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The concepts for this exercise are summarized below.
For a complete discussion, read the text.
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THEOREM: solving absolute value equations
Let
x∈ℝ ,
and let
k≥0 .
Then,
|x|=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, " x=± k
" is a shorthand for
" x=k or
x=-k ".
When you see a sentence of the form
" |x|=k
", here's what you should do:
- Check that k is a nonnegative number.
- The symbol |x| represents the distance between x and 0 .
- Thus, you want the numbers x , whose distance from
0 is k .
- You can walk from 0 in two directions: to the right, or to the left.
Thus, there are two numbers whose distance from 0 is the nonnegative number k :
k and
-k .
- Thus, |x|=k
is equivalent to
x=± k
.
However, the power of the tool
" |x|=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
⇔ 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 equation) |
| 2-3x=
±7 |
(use the theorem) |
| 2-3x=
7 or
2-3x=-
7 |
(expand the plus/minus) |
| -3x=5
or -3x=-9
|
(subtract 2 from both sides of both equations) |
| x=-5
3 or
x=3 |
(divide both sides of both equations by -3 ) |
Solve:
5-2|3
-4x|=-
7
Solution: Write a nice, clean list of equivalent sentences:
| 5-2|3
-4x|=-
7 |
(original equation) |
|
-2|3-
4x|=-12
|
(subtract 5 from both sides) |
| |3-4x
|=6 |
(divide both sides by -2 ) |
|
3-4x=
±6 |
(use the theorem) |
| 3-4x=
6 or
3-4x=
-6 |
(expand the plus/minus) |
| -4x=3
or
-4x=-9
|
(subtract 3 from both sides of both equations) |
| x=-3
4 or
x=9
4 |
(divide both sides of both equations by -3 ) |
Solve: |3x+1
|=-5
Solution: always false
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