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Using Calculators to Solve Inequalities (Cambridge (CIE) IGCSE International Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Using Calculators to Solve Inequalities

How do I solve inequalities using my graphic display calculator?

  • You can use a graphic display calculator to solve inequalities of the form

    • straight f subscript 1 open parentheses x close parentheses greater than straight f subscript 2 open parentheses x close parentheses or straight f subscript 1 open parentheses x close parentheses less than straight f subscript 2 open parentheses x close parentheses

    • straight f subscript 1 open parentheses x close parentheses greater or equal than straight f subscript 2 open parentheses x close parentheses or straight f subscript 1 open parentheses x close parentheses less or equal than straight f subscript 2 open parentheses x close parentheses

  • First, turn the inequality into an equation and solve straight f subscript 1 open parentheses x close parentheses equals straight f subscript 2 open parentheses x close parentheses

    • Read the revision note on Using Calculators to Solve Equations to see how

    • The solutions to the equation straight f subscript 1 open parentheses x close parentheses equals straight f subscript 2 open parentheses x close parentheses are the x-coordinates of the points of intersection of the two graphs y equals straight f subscript 1 open parentheses x close parentheses and y equals straight f subscript 2 open parentheses x close parentheses

  • Then look at the regions (areas) enclosed between the two graphs and decide which curve is the top curve and which is the bottom curve

    • e.g. The solution to straight f subscript 1 open parentheses x close parentheses greater than straight f subscript 2 open parentheses x close parentheses are the ranges of x-values for which y equals straight f subscript 1 open parentheses x close parentheses is the top curve and y equals straight f subscript 2 open parentheses x close parentheses is the bottom curve

      • The top curve is greater than the bottom curve

    • Examples of different regions, R, are shown below

    A diagram showing whether a curve is above a line or the line is above a curve

    • For example, to solve x squared plus 3 x plus 1 greater than 2 x plus 1 using a graphical method

      • First solve the "=" equation to get x equals negative 1 and x equals 0

      • Then plot on your calculator and look for any regions where the quadratic is above (greater than) the line (quadratic on top, line on bottom)

      • This happens to the left of x equals negative 1 and to the right of x equals 0

      • The solution is x less than negative 1 or x greater than 0

Points of intersection between a curve and a line

Examiner Tips and Tricks

Remember to give your solutions as ranges of x values, using strict inequalities if the question uses strict inequalities!

Worked Example

Use a graphical method to solve the inequality x cubed plus 2 x cubed plus 1 less or equal than x plus 2, giving your answers correct to 1 decimal place and showing a sketch of the graph.

Start a new graph on your calculator and enter straight f subscript 1 open parentheses x close parentheses equals x cubed plus 2 x squared plus 1 to generate the graph
Add on the second graph of straight f subscript 2 open parentheses x close parentheses equals x plus 2

Select the option to "analyze" the graph (this may be labelled as G-Solv)
Choose the option "intersections" to find the coordinates of the 3 points of intersection

The solutions to the equation are the x-coordinates of the points of intersection

Drawing a cubic graph and a straight line graph on a graphic display calculator to find the points of intersection

The solutions to the equation x cubed plus 2 x cubed plus 1 equals x plus 2 are
x equals negative 2.2, x equals negative 0.6 or x equals 0.8 to 1 d.p.

The inequality in the question is "cubic graph less or equal than straight line graph"
Look at regions where the cubic is the bottom curve and the line is above

left of x = -2.2 and between x = -0.6 and x = 0.8

The solutions are the ranges of x-values over which these regions lie
Write your final answer using non-strict inequalities (to match the question)

bold italic x bold less or equal than bold minus bold 2 bold. bold 2 or bold minus bold 0 bold. bold 6 bold less or equal than bold italic x bold less or equal than bold 0 bold. bold 8 to 1 d.p.

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    Previous:Using Calculators to Solve EquationsNext:Representing Inequalities as Regions
    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.