Inequalities

INEQUALITIES

Inequality- Compares two numbers or expressions.
Solving an inequality is similar to the process of solving an equation, but there are a few exceptions.
Such as: When both sides of an inequality are divided or multiplied by a negative number, the direction of the inequality symbol must be reversed.

Properties of Inequalities

1. Transitive Property

      

2. Addition of Inequalities

    

3. Addition of a Constant

  

4. Multiplying by a Constant

 ,

 , 

A.  Linear Inequalities

              add 7 to each side

              subtract 3x from each side

                        combine like terms

                             divide each side by 2

                            Solution set consists of all real numbers that are greater than 8.

B.  Polynomial Inequalities

                                            

                               factor

                               Treat as equation

                  Split

                       Solve.     Critical Values: X-intercepts. Only places where the function could change  from 

being negative to positive or vice versa.

Number Line

Test Intervals:       

Pick a test value for each test interval. Plug value into original inequality to see if answer is negative or positive.

Test Interval:

Test Value:  

Result: 

Test Interval:  

Test Value: 

Result: 

Test Interval: 

Test Value: 

Result: 

               SOLUTION

                            INTERVAL NOTATION

Graphing can verify your solution. If we graphed this inequality it would look like the parabola below. We can see from the graph that the interval of -2 to 3 is below the x-axis and is thus less than 0. 

A shortcut:

1.  Solve for the x-intercepts, places where the function changes from being negative to positive and 

vice versa :   -2, 3

2. Examine the leading coefficient of your original quadratic function:  +1 implies parabola opens upward

3. Sketch a graph of the function

:

4. Examine Inequality and Graph:   Original inequality stated that solutions were < 0.     On the graph the negative solutions are in the interval   . So the solution can be any real number within the interval of .

If you would like another example of solving a polynomial inequality, here's a neat video. 

http://www.youtube.com/watch?v=4lE3PXm9uWQ

Hope this was helpful!

-Leah