Rational Inequalities
      The graph of the translated equilateral (or rectangular) hyperbola
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.
Example:  Find the solutions of the inequality    
Solution:  The solution set of the given rational inequality includes all numbers x which make the inequality less then or equal to 0, or which make the sign of the rational expression to be negative or 0.
A rational expression is negative if the numerator and the denominator have different signs, and the rational expression equals 0 when its numerator is equal to 0 that is,
therefore, we have to solve two simultaneous inequalities.
We graph the numerator and the denominator in the same coordinate system to find all points of the x-axis that satisfy given inequality.
The zero points of the numerator and the denominator divide the x-axis into four intervals at which given rational expression changes sign.
x2 + 2x - 3 = 0,  a = 1, b = 2 and c = -3
 
The solutions of the two pairs of the simultaneous inequalities are intersections of sets of their partial solutions,
as is shown below.
Therefore, the solution set is (- oo, - 3] U (- 2, 1].
Intermediate algebra contents
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