  Solving and graphing systems of linear inequalities in two variables
Rational Inequalities Method of solving rational inequalities The graph of the translated equilateral (or rectangular) hyperbola
Systems of Linear Inequalities
The solution of the system of simultaneous inequalities is the intersection of sets of the individual solutions.
Example:  Solve and graph the solution of the given system of simultaneous inequalities.
 Solution:  the solution of the system is the open interval (3, + oo ).
Solving and graphing systems of linear inequalities in two variables
The set of points whose coordinates (x, y) satisfy the inequality  ax+ by + c > 0 is a half-plane of a Cartesian plane.
Example:  Solve and graph the solution of the given system of linear inequalities in two variables.
 Solution: Plug the coordinates of the origin  Thus, non shaded area of the Cartesian plane where lies the origin, bounded by given lines, is the solution of the system of inequalities.
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 >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 ).
 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 