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ALGEBRA
 :: 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
Method of solving rational inequalities

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.

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).

Example: Solve the rational inequality

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. Therefore, we have to solve the 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, 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.

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].
 :: Absolute value inequalities

Graphical interpretation of the definition of the absolute value of a function = f (x) will help us solve linear inequality with absolute value. The definition for the absolute value of a function is given by

  
Solving linear inequalities with absolute value

Thus, for values of x for which  f (x) is nonnegative, the graph of | f (x)| is the same as that of f (x).

For values of x for which f (x) is negative, the graph of | f (x) | is a reflection of the graph of f (x) on the x-axis. That is, the graph of  = - f (x) is obtained by reflecting the graph of = f (x) across the x-axis.
Example: Solve the absolute value inequality  | x - 2 | < 3.

Solution:  Graphical interpretation of the given inequality will help us find values of x for which left side of the inequality is less then or equal to 3.

For values of x for which y is nonnegative, the graph of | y | is the same as that of  = x - 2.
For values of x for which y is negative, the graph of | y | is a reflection of the graph of y across the x axis.
Since the graph of  = x - 2 has y negative on the interval (- oo, 2 ) it is this part of the graph that has to be reflected on the x axis.
The graph shows that values of x from the closed interval [-1, 5] satisfy the given inequality.
The same result can be obtained algebraically by solving the compound inequality.
the interval notation x [-1, 5]
 :: Systems of linear inequalities
Solving and graphing systems of linear inequalities
The solution of the system of simultaneous inequalities is the intersection of sets of 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 ).
 :: Systems of linear inequalities in two variables
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
 :: Quadratic inequalities
To solve a quadratic inequality we examine signs of the equivalent quadratic function.
The x-intercepts or roots are the points where a quadratic function changes the sign. The x-intercepts determine the three intervals on the x-axis in which the function is above or under the x-axis, that is, where the function is positive or negative.
Graphic solution of quadratic inequalities
Example: Solve the inequality  - x2 + 2x + 3 0.
Solution:  Solve the quadratic equation  ax2 + bx + c = 0 to get the boundary points.
The zeroes or roots of equivalent function (see the graph below) are the endpoints of the intervals and are included in the solution.
The turning point  V(x0, y0),    V(1, 4)
The roots of,    - x2 + 2x + 3 = 0
Solution:
 
 
 
 
 
 
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