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Solve the inequality |2-3x| < 11

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2 Answers

|2-3x| < 11
 
3|x-2/3| < 11
 
|x-2/3| < 11/3
 
The distance of x from 2/3 is less than 11/3.
 
<---O========+=========O--->
    -9/3                2/3                  13/3
 
-3 < x < 13/3
Hi Kay;
In the title, you explain this is <.
In the description, you explain this is ≤.
I will use the first...
|2-3x| < 11
Absolute value is always positive.
(2-3x) can be either...
+(2-3x) or -(2-3x).
When the absolute value of either is taken, it has the same result.
-(2-3x)<11
-2+3x<11
Let's add 2 to both sides...
-2+2+3x<11+2
3x<13
Divide both sides by 3...
(3x)/3<13/3
x<13/3
 
+(2-3x)<11
2-3x<11
Let's subtract 2 from both sides...
2-2-3x<11-2
-3x<9
Divide both sides by -3.  Because this is a negative number < will switch to >...
(-3x)/-3>9/-3
x>-3
 
-3<x<13/3
Let's verify by plugging-in a few points within this domain...
|2-3x| < 11
 
x=0
|2-[(3)(0)]|<11
|2|<11
2<11-----YES
 
x=-1
|2-3x| < 11
|2-[(3)(-1)]|<11
|2+3|<11
5<11---YES
 
x=1
|2-3x| < 11
|2-[(3)(1)]|<11
|2-3|<11
-1<11-------YES
 
Let's verify by plugging-in a few points outside of this domain...
x=-4
|2-3x| < 11
|2-[(3)(-4)]|<11
|2+12|<11
13<11-----NO
 
x=5
|2-3x| < 11
|2-[(3)(5)]|<11
|2-15|<11
|-13|<11
13<11-------NO