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
