Linear Programming Including Inequalities: MYP 5 Mathematics Notes

What is Linear Programming?

Linear programming is a powerful mathematical method used to find the best possible outcome in a situation where there are limitations or restrictions. The word linear tells us that all the relationships involved are straight lines when drawn on a graph, and programming here means planning or optimising , not computer coding!

Linear Programming: A mathematical technique used to find the maximum or minimum value of a quantity (called the objective function) subject to a set of restrictions called constraints, all expressed as linear inequalities.

Linear programming is used extensively in the real world to solve problems like:

A factory deciding how many of two products to make in order to maximise profit
A dietitian planning meals to minimise cost while meeting nutritional needs
An airline scheduling crew to minimise expenses while satisfying regulations
A farmer deciding how much of each crop to plant to maximise yield

Analogy

Imagine you are running a small bakery. You can bake cakes or cookies, but you only have a limited amount of flour, butter, and time. Linear programming helps you figure out exactly how many of each to bake so that you make the most money possible , without breaking those resource limits.

In MYP Sciences, linear programming connects the Thinking with Models unit beautifully , the graphical model we build (a shaded feasible region on a coordinate plane) represents all the possible solutions, and we then identify the optimal one.

Reviewing Inequalities

Before tackling linear programming, you need to be confident with inequalities , mathematical statements that compare two expressions using symbols other than equals.

Inequality: A mathematical statement that shows the relationship between two expressions that are not necessarily equal, using the symbols $<$ , $>$ , $\leq$ , or $\geq$ .

The four inequality symbols and their meanings:

Symbol	Meaning	Example
$<$	strictly less than	$x < 5$
$>$	strictly greater than	$y > 2$
$\leq$	less than or equal to	$x \leq 10$
$\geq$	greater than or equal to	$y \geq 0$

Note

In most real-life linear programming problems, we use $\leq$ or $\geq$ rather than strict inequalities, because a constraint like "you have at most 100 kg of flour" includes using exactly 100 kg as a valid option.

Solving a simple inequality works almost exactly like solving an equation, with one critical rule:

Warning

Flip the inequality sign whenever you multiply or divide both sides by a negative number.

For example: $- 2 x \leq 8$ → divide both sides by $- 2$ → $x \geq - 4$ (sign flips!)

Example

Solve: $3 x + 4 \leq 19$

Step 1: Subtract 4 from both sides: $3 x \leq 15$

Step 2: Divide both sides by 3: $x \leq 5$

Interpretation: Any value of $x$ that is 5 or less satisfies this inequality.

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