Rational Inequalities
      Method of solving rational inequalities
      The graph of the translated equilateral (or rectangular) hyperbola
Rational Inequalities
A rational inequality can be written in one of the following standard forms: 
P/Q > 0  or  P/Q < 0  (or P/Q > 0 or P/Q < 0 ), where Q is not 0
The sign of the rational expression P/Q, where P and Q are polynomials, depends on the signs of P and Q.
As the signs of the polynomials change at the zeros, to solve a rational inequality we should find the zeros of both P and Q first and then we can determine the intervals of the independent variable that satisfy given rational inequality.
Method of solving rational inequalities
The first step to solve a rational inequality is to get a single rational inequality on the left side of the inequality sign and have zero on the right side of the inequality sign.
The next step is to factor the numerator and denominator and find the values of x that make these factors equal to 0 to find critical points (boundaries, endpoints of intervals).
Note that, by setting the numerator to 0 we get the zero points of the given rational expression, but by setting the denominator to 0 we get points at which the rational expression or function is undefined (i.e., when plugged into the expression give division by zero).
The rational functions are not defined at the zeros of the denominator. Therefore, they have breaks or vertical asymptotes at these points, and this is why these points cannot be included into the solution of the rational inequality.
Example:  Solve the rational inequality and draw the graph of the rational function.
Solution:  The solution set of the given rational inequality includes all numbers x which make the inequality greater then or equal to 0, or which make the sign of the rational expression to be positive or 0.
A rational expression is positive if both the numerator and the denominator are positive or if both are negative, and the rational expression equals 0 when its numerator is equal to 0 that is
therefore, we have to solve two simultaneous inequalities:
The solutions represented on the number line are shown below. 
Thus, the solution set of the given inequality written in the interval notation is (- oo, -1) U [2, oo ).
The graph of the translated equilateral or rectangular hyperbola
The graph of the given rational function is translated equilateral (or rectangular) hyperbola.
A rational function of the form   can be rewritten into
   
  where the vertical asymptote, the horizontal asymptote  
  and the parameter  
Therefore, values of the vertical and the horizontal asymptote correspond to the coordinates of the horizontal and the vertical translation of the source equilateral hyperbola  y = k/x, respectively.
Thus, given rational function   where, a = 1, b = -2  and  c = 1, d = 1
  has the vertical asymptote    
  the horizontal asymptote    
  and the parameter    
Therefore, its source function is the equilateral or rectangular hyperbola    
The graph of given rational function is shown below, (see the above example of the rational inequality).
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