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♥ INTERNET EXPLORER 6.0 and above, with MathPlayer installed.♥
SOLVING ABSOLUTE VALUE SENTENCES, ALL TYPES
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MathPlayer is only required for the concept discussion; it is not required for the web exercise.
This web exercise mixes up problems from the previous three web exercises:
Solving Absolute Value Equations
Solving Absolute Value Inequalities Involving "Less Than"
Solving Absolute Value Inequalities Involving "Greater Than"
Visit these earlier sections for a thorough discussion of the concepts.
THEOREM: solving absolute value sentences
Let
x∈ℝ ,
and let
k≥0 .
Then,
|x|=k
⇔ x=±
k
|x|<k
⇔ -k<x<k
|x|≤k
⇔ -k≤x≤k
|x|>k
⇔
x>k
or x<-k
|x|≥k
⇔
x≥k
or x≤-k
|
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:
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:
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) |
|
-6x+7
>3 or
-6x+
7<-3
|
(use the theorem) |
| -6x>-
4 or
-6x<-10
|
(subtract 7 ) |
| x<4
6 or
x>
106 |
(divide by -6 ; change the direction of the inequality symbols) |
| x<2
3 or
x>
53 |
(write fractions in simplest form) |
Solve:
|5-2x
|>-3
Solution: always true
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.