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FM Single Variable

2.3 Linear Programming

Note: these notes cover the formulation and understanding of linear programming questions, methods for determining solutions to these problems are covered in notes for 2.4 Graphical Method for Solving Linear Programming Problems and 2.5 Integer Solutions for Linear Programming Problems.

Linear Programming and its Practical Applications

  • Linear programming describes a process whereby we wish to maximise the value of a particular function, known as the objective function.
  • The objective function is based on a number of variables, known as the decision variables. In the scope of Further Maths, we will analyse situations with two decision variables.
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2.2 Graphs of Linear Inequalities

Graphs of Single Variable Inequalities

  • Inequalities can be represented graphically as a shaded area.
  • For single variable inequalities, the shaded area is enclosed by two vertical (for x inequalities) or horizontal (for y inequalities) lines.


Below is a graph representing the inequality -2 \leq x \leq 1:

Picture 1

Graphs of Two-Variable Inequalities

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2.1 Linear Inequalities

Single Variable Inequalities

  • Inequalities provide a range of values that a variable can possess.
  • Inequalities of the form x>a or x<a where x is a variable and a is a constant tell us that x is greater than a or less than a, respectively.
  • Adding a line beneath the inequality symbols indicates that the variable can also be equal to the value it is being equated to e.g. x \geq a means x is greater than or equal to a.
  • Inequalities can also be used to represent a finite range of values using the form:


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