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

Solving inequalities graphically

How do I solve inequalities graphically?

• Consider the inequality , where and are functions of

  • Subtract from both sides to get zero on the right-hand side

    • 

• Solve the equation to find the -intercepts of the graph

• Sketch the graph of

  • Use your GDC to help

  • Label its -intercepts

  • The solutions are the range(s) of values of for which the curve is

    • below the -axis

    • as

  • Present your solutions as inequalities

    • e.g.

• If the inequality in the question had been reversed,

  • then

    • solutions are the values of where the curve is above the -axis

When do I flip the inequality sign?

• Remember to flip the sign of the inequality when you multiply or divide both sides by a negative number

  • e.g. becomes when dividing both sides by

• Never multiply or divide both sides by a variable as this could be positive or negative

  • e.g. if then is not always true

    • e.g. satisfies but not

    • you get

  • This means you can only multiply both sides by a terms that are always positive

    • Such as 

• Taking reciprocals of positive values reverses the inequality

  • If then

• Taking logarithms when the base is reverses the inequality

  • 

• The safest way to rearrange is simply to add and subtract terms to both sides

  • e.g. becomes