what are the rules for absolute values and inequalities in math? for example...?

Question

what is value of x? abs of (x+2)/ abs of (x-1) = 3

what is the soln for this inequality 4/ (x-2) < 2


what are the rules for soling any abs value equations and what are the rules for solving inequalities < & >?

Answer 1

|x + 2|/|x - 1| = 3
|x + 2| = 3 * |x - 1|
|x + 2| = |3x - 3|

There are four cases here.
x + 2 = 3x - 3
-(x + 2) = 3x - 3
x + 2 = -(3x - 3)
-(x + 2) = -(3x - 3)
But these resolve into two case, actually.

x + 2 = 3x - 3
5 = 2x
x = 2.5

-(x + 2) = 3x - 3
-x - 2 = 3x - 3
1 = 4x
x = .25

x + 2 = -(3x - 3)
x + 2 = -3x + 3
4x = 1
x = .25

-(x + 2) = -(3x - 3)
-x - 2 = -3x + 3
2x = 5
x = 2.5



The important thing about inequalities is that when you multiply by a negative number, you have to flip the inequality sign.

4/(x-2) < 2
4 < 2(x-2)
4/2 < x-2
2 < x - 2
4 < x, and x <> 2

x <> 2 because then the equation on the left is undefined.

Answer 2

converting fraction - divide!