Quadratic Inequalities (DP IB Analysis & Approaches (AA)): Revision Note

Quadratic Inequalities

What affects the inequality sign when rearranging a quadratic inequality?

• The inequality sign is unchanged by

  • Adding or subtracting a term to both sides

    • E.g. becomes

  • Multiplying or dividing both sides by a positive term

    • E.g. becomes

• The inequality sign flips (< changes to >) when

  • Multiplying or dividing both sides by a negative term

    • E.g. becomes

How do I solve a quadratic inequality?

• STEP 1

  Rearrange the inequality into quadratic form with a positive squared term and with zero on one side

  • E.g.

• STEP 2

  Find the roots of the quadratic equation

  • Solve to get and where

• STEP 3
  Sketch a graph of the quadratic and label the roots

  • As the squared term is positive it will be concave up, i.e. U-shaped

• STEP 4
  Identify the region on the graph which satisfies the inequality

  • If you want the graph to be above the -axis then the region will be the two intervals outside of the two roots

    • E.g. for , the solution is or

    • and for , the solution is or

  • If you want the graph to be below the -axis then choose the region to be the interval between the two roots

    • E.g. for , the solution is

    • and for , the solution is

How do I solve a quadratic inequality of the form (x-h)²<n or (x-h)²>n?

• The safest way is to expand and rearrange, then follow the steps above

• A common mistake is writing  or 

  • This is NOT correct!

• The correct solution to is

  • which can be written as 

  • The final solution is 

• The correct solution to is

  • which can be written as  or 

  • The final solution is or