Solve Rational Inequalities

To solve rational inequalities, it helps by rewriting with the right side equal to zero. A number line can be used to consider intervals. Then pick a number of each interval to determine the sign of the factors. 

For example, x/x+1 < 2x/x-2

x/x+1 - 2x/x-2 <0


x(x-2) - 2x(x+1) < 0
    (x-2)(x+1)


x^2-2x-2x^2-2x < 0
    (x-2)(x+1)

-x^2 - 4x   < 0

(x-2)(x+1)

-x ( x+4 )   < 0

(x-2)(x+1)


Restriction of x:
x cannot be -1 and 2.


1. List all the factors.

2. Use number line/table to help to consider each intervals.

3. Pick a number and test the value. 

Factors / Intervals	x < -4	-4 < x < -1	-1 < x < 0	0 < x < 2	x > 2
Test Values	-5	-2	-0.5	1	3
-x	+	+	+	-	-
x+4	-	+	+	+	+
x-2	-	-	-	-	+
x+1	-	-	+	+	+
	-	+	-	+	-

x < -4, -1 < x < 0 and x > 2 are negative or less than zero. 

Hence the solution of x/x+1 - 2x/x-2 <0 are x < -4, -1 < x < 0 and x > 2.

Source: 

McGraw-Hill Ryerson Advanced Functions 12

A simple example from this video :)