Inequalities and Linear Programming

Linear inequalities

Solve like an equation, with one rule: multiplying or dividing by a negative number reverses the sign.

$$3 - 2x > 7 \Rightarrow -2x > 4 \Rightarrow x < -2$$

Compound inequalities:

• "and": both must hold, so take the overlap.
• "or": either may hold, so take everything covered by either.

Quadratic inequalities

1. Move everything to one side and factorise.
2. Sketch $y = (x - p)(x - q)$, a U-shape cutting the $x$-axis at $p$ and $q$ ($p < q$).
3. Read off:

Inequality	Solution
$(x - p)(x - q) < 0$	$p < x < q$ (between the roots)
$(x - p)(x - q) > 0$	$x < p$ or $x > q$ (outside the roots)

Common mistake: $x^2 \geq 9$ is not $x \geq \pm 3$. It is $x \leq -3$ or $x \geq 3$.

Linear programming

1. Define the variables, e.g. $x$ trays of cookies and $y$ trays of cakes.
2. Write each constraint as an inequality, including $x \geq 0$ and $y \geq 0$.
3. Shade the region that satisfies all of them. DSE calls it "the shaded region (including the boundary)".
4. The greatest or least value of $P = ax + by$ occurs at a vertex of the region. Evaluate $P$ at each vertex and compare.

If the variables must be whole numbers (people, trays, buses), check that the best vertex has integer coordinates; otherwise test the nearby integer points inside the region.

Exam tip: to find a vertex where two boundary lines meet, solve their equations simultaneously. Show the values of $P$ at all vertices in a table.