Notes on using Ineq.xls

This spreadsheet allows students to explore the behaviour of the solution of a quadratic inequality.

The approach is graphical.

It could be used as an introduction to or as a reinforcement of the theory of quadratic inequalities.

It is possible for the inequality to reduce to a linear one and students should be encouraged to take account of this when answering the questions.

Background: Theory of quadratic equations and the graphical solution of linear and quadratic inequalities.

Questions for students:

When considering solutions to inequalities of the form

AX² + BX + C >0 or AX² + BX + C <0

• What types of solution to the inequality are possible? Are there any that the spreadsheet does not show or deal with?

• Give some examples of inequalities which have no solution. How many different sorts can you find?

• Give some examples of inequalities which have all real numbers as their solution. How many different sorts can you find?

• Under what circumstances (in terms of conditions on the values of A,B and C) will there be no solution to the inequality? Make sure you have considered all possible ways in which this could happen.

• Under what circumstances will the solution be all real numbers? Make sure you have considered all possible ways in which this could happen.

• Give some examples of inequalities which have solutions of the form x<n.

• Under what circumstances will there be a solution to the inequality of the form x<n ?

• Give some examples of inequalities which have solutions of the form x>n.

• Under what circumstances will there be a solution to the inequality of the form x>n ?

• Give some examples of inequalities which have solutions of the form m<x<n.

• Under what circumstances will there be a solution to the inequality of the form m<x<n ?

• Give some examples of inequalities which have solutions of the form x<m or x>n.

• Under what circumstances will there be a solution to the inequality of the form x<m or x>n?

• Pair up with another student. Give each other four different inequalities of the form
  AX² + BX + C >0 or AX² + BX + C <0 and see if you can predict the type of solution for each one.
  Use the spreadsheet to check (so use values of A,B and C between –20 and 20).