Learning Goal

Syllabus Requirement

Students need to learn the solution of linear and quadratic inequalities, with the examples below highlighted in particular:

  •      ??ax + b > cx + d??
  •      ??px^{2} + qx + r \geq 0??
  •      ??px^{2} + qx + r < ax + b??

Study Notes

There are 2 types of inequalities that you should be able to solve:
Linear inequalities, which involve linear functions of a variable.
Quadratic inequalities, that involve quadratic functions of a variable.

The Methods for Solving Inequalities

Linear Inequalities
These can be solved in the same way as linear equations, with the exception that: any division or multiplication by a negative number, reverses the direction of the inequality.
For example, ??< ?? would become ??>?? and vice versa.

Quadratic Inequalities: Graphical Approach
The first method of solving quadratic inequalities, is to find the roots of the quadratic function, then to sketch the graph to find the range of values of ??x?? that satisfy the inequality.

Quadratic Inequalities: Algebraic Approach
The second method of solving quadratic inequalities, is to find the roots of the quadratic function, as normal algebraically. The problem then is to identify which range(s) of ??x?? satisfy the inequality. There will be a maximum of 3 ranges of ??x?? and each one can be tested by picking a value in that range and checking whether it does or does not satisfy the inequality.

Worked Examples

We'll now run through some examples that cover each of the methods, note that we use the same quadratic function for both examples relating to quadratic functions, to show that they both produce the same answer:

Linear Inequalities

Example 1


Solve ?? -5x - 11\leq 4??.
Show Solution

Quadratic Inequalities: Graphical Approach

Example 2


Find the range of values for which ??x^{2} - 4x - 12 < 0??.
Show Solution

Quadratic Inequalities: Analytical Approach

Example 3


Find the range of values for which ??x^{2} - 4x - 12 < 0??.
Show Solution

Exam Questions

The table below contains every exam question that has been asked on this topic, this includes normal papers from both January and June sittings, International papers and Specimen papers.

To see an exam question and solution simply click on the load question icon (), the question will appear below the table and the solution can be shown by clicking the "Show Solution" button that also appears.

The full exam paper and mark schemes are also available for download by clicking on the download icons in each row of the table ().

Exam Board Subject Paper Year Month Module Question
Parts Total
Edexcel Maths Standard 2005 January Core 1 10 n/a 12
Edexcel Maths Standard 2005 June Core 1 6 n/a 8
Edexcel Maths Standard 2006 June Core 1 2 n/a 4
Edexcel Maths Standard 2009 June Core 1 4 n/a 7
Edexcel Maths Standard 2010 June Core 1 3 n/a 6
Edexcel Maths Standard 2012 January Core 1 3 n/a 6
Edexcel Maths Standard 2013 June Core 1 5 n/a 6
Edexcel Maths International 2013 June Core 1 8 n/a 8

Exam Tips

  1. Typically questions will test whether students can deal with the effect of dividing or multiplying an inequality by a negative amount (i.e. the switching of inequality signs).

  2. Most questions involving quadratic inequalities will necessitate that students are familiar with the algebraic or graphical approach of finding the range after simplifying the inequality. This is such a common component of these questions that you should be familiar with both methods and able to execute them accurately and quickly.