Inequalities — When "Equal" Isn't the Question

Inequalities

Pure Mathematics 1 · Algebra and Functions · Lesson 5

Driving Question: Most of life isn't "exactly equal" — it's "at least", "no more than", "somewhere between". How do we solve a question whose answer is a range rather than a point?

An equation asks "where are two things equal?" and usually answers with a handful of exact points. An inequality asks a softer, more realistic question: "where is one thing bigger than another?" Its answer is rarely a single value — it's a whole region of the number line. This lesson is about finding those regions, drawing them honestly, and — for quadratics — seeing why the answer is sometimes an interval and sometimes two pieces flying apart.

🧠 The Knowing World

Linear inequalities

You solve a linear inequality almost exactly as you would an equation — add, subtract, multiply, divide to isolate the variable. There is just one rule that bites:

Flip the sign when you multiply or divide both sides by a negative number. If −2x < 6, dividing by −2 gives x > −3 — the < becomes >. (Why? Multiplying by a negative reflects every number through zero, which reverses their order.)

Example. Solve 3x − 5 ≤ 7.
Add 5: 3x ≤ 12. Divide by 3 (positive, so no flip): x ≤ 4.
Solution set: all x with x ≤ 4.

On a number line we mark the boundary with an open circle for < or > (the boundary is not included) and a closed circle for ≤ or ≥ (it is included), then shade the region that satisfies the inequality.

Quadratic inequalities

Here is the key idea, and it is worth saying slowly: to solve a quadratic inequality, first find where the quadratic equals zero (its roots), then decide which side(s) of those roots make the inequality true. The roots are the only places the expression can change sign, so they cut the number line into regions — and within each region the sign never changes.

Example. Solve x² − x − 6 > 0.
Factorise: (x − 3)(x + 2) = 0 at x = 3 and x = −2.
The parabola opens upwards, so it is above the x-axis (positive) outside the roots.
Solution: x < −2 or x > 3. (Two separate regions.)

Same quadratic, flipped. Solve x² − x − 6 < 0.
Same roots, but now we want where the parabola is below the axis (negative) — between the roots.
Solution: −2 < x < 3. (One interval.)

A sketch beats a memorised rule every time. Draw the parabola, mark the roots, see whether you want the part above or below the axis, and read the region straight off the picture. "Above the axis → outside the roots; below the axis → between the roots" is something you can see, not just recall — provided the x² coefficient is positive (if it's negative, the parabola opens downward and the regions swap).

Seed of Change: Solve linear and quadratic inequalities, including representing the solution set on a number line.

🌍 The Human Story

The symbols < and > are surprisingly modern. They were introduced by the English mathematician Thomas Harriot, whose work was published in 1631 — long after the equals sign, which Robert Recorde had devised back in 1557. For a remarkably long stretch of mathematical history, then, people could write "these two things are equal" with a tidy symbol but had no quick way to write "this one is larger". The idea of ordering — of more and less — had to wait for its own notation.

That gap is telling, because ordering is arguably the more primitive idea. Long before anyone could solve an equation, people compared: which harvest was bigger, which portion was fairer, which path was shorter. Inequalities are the mathematics of comparison, and comparison is one of the oldest things human beings do.

They are also the mathematics of the real world in a way exact equations often are not. A bridge must carry at least a certain load. A medicine must keep a concentration between a floor (too little does nothing) and a ceiling (too much is dangerous). A budget must come in at or under a limit. Engineers, doctors, and economists spend far more of their time with inequalities than with equations, because the questions that actually matter are usually about safe ranges and acceptable limits rather than single perfect values.

A fairness reflection: an enormous amount of justice is, at its heart, about inequalities — thresholds of need, floors below which no one should fall, ceilings on what may be taken. The mathematics of "at least" and "no more than" is quietly the mathematics of how a society decides what is enough.

🎨 The Expressive Self

Choose one and make the idea your own:

In words: Explain to someone why solving x² − x − 6 > 0 gives two separate regions, while changing the > to a < gives one region in the middle. Use a picture of the parabola, not a rule.
As a sketch: Draw one upward parabola crossing the x-axis at two points. Shade, in one colour, the x-values where it's above the axis; in another, where it's below. Label each with the inequality it solves.
As a story: Describe a real situation governed by "between two limits" — a thermostat, a safe dosage, a comfortable speed — and write the inequality that captures it.
Spot the trap: Write a short note to a future student warning them about the single most common inequality mistake (forgetting to flip the sign when dividing by a negative). Make it memorable.

🛠️ The Designed World

Two explorers below. Stage 1 lets you build a linear inequality and see its solution set drawn honestly on a number line, open or closed circle and all. Stage 2 is the heart of the topic: drag the parabola and choose the inequality, and watch the satisfying region of the x-axis light up — so you can see why "above the axis" means "outside the roots" and "below" means "between" them.

Stage 1 · Linear inequality on a number line

Build ax + b (relation) 0, then read its solution set off the number line. Watch the circle switch between open and closed, and watch the sign flip when a is negative.

Coefficient a = 2

Constant b = -6

Relation:

Set a = 2, b = −6, relation ≤. Read the solution. Now switch a to −2 without changing anything else. What happens to the inequality sign, and why?
Find a setting where the solution is x > 0 exactly. What does the open circle at zero mean — is x = 0 part of the solution?
Set a = 0. The "inequality" no longer depends on x. When is it true for all x, and when for none? Explain what the explorer shows.

Stage 2 · Quadratic inequality — see the region

The parabola is y = x² + px + q. Drag p and q to move its roots, choose the inequality, and the x-values that satisfy it are highlighted on the axis. Notice when the solution is one interval, when it's two outer regions, and when it's empty or everything.

p = -1

q = -6

Inequality:

Try dragging to find nice roots yourself first, then snap to tidy them.

Parabola Roots Solution region on the x-axis

Set the inequality to y > 0 and find roots at x = −2 and x = 3 (use snap). Read the solution. Now switch to y < 0 without moving anything. How does the region change — and why is one "outside" and the other "between"?
Drag q upward until the parabola just touches the axis (one repeated root). What is the solution set of y > 0 there? And of y ≥ 0? (The difference between > and ≥ matters most exactly here.)
Drag q higher still, so the parabola sits entirely above the axis. What is the solution of y > 0 now? Of y < 0? Connect your answer to the discriminant.
Productive friction: the explorer always draws an upward parabola (the x² coefficient is +1). Sketch by hand what would change if it opened downward. Which regions would swap, and why?

💛 The Living Body

Inequalities trip up a lot of capable people, and almost always at the same two spots: forgetting to flip the sign when dividing by a negative, and muddling "between the roots" with "outside the roots". If either of those has caught you, you are in very normal company — these aren't signs of weakness, they're signs that the topic has two genuine pinch-points worth slowing down for.

Notice, gently:

When you divided by a negative, did you catch the sign flip in the moment, or only on checking? Building the habit of pausing at every negative divide is worth more than speed.
For the quadratic case, did the picture help? If "draw the parabola, see the region" felt more secure than any rule, trust that — a method you can see is a method you can rebuild when you're tired or anxious.
Which felt steadier: the number line, or the parabola? There's no right answer. Knowing your stronger entry point is a tool for the next time the topic appears.

If you noticed frustration at any point, what helped it settle — a sketch, a pause, the explorer, re-reading slowly? That's a strategy worth keeping.

🛠 Final Task

Build an "Inequality Field Guide." Produce a short illustrated reference (handwritten, typed, or recorded) containing all of the following:

A linear inequality requiring a sign flip, solved fully, with its solution set drawn correctly on a number line (open/closed circle shown deliberately).
A quadratic inequality of the "> 0" type, solved with a sketch, giving a two-region ("or") answer.
A quadratic inequality of the "< 0" type using the same quadratic, giving a single-interval answer — placed next to the first so the contrast is visible.
A short paragraph (3–4 sentences) explaining, in your own words and using a sketch, why "above the axis" means outside the roots and "below" means between them — and what changes if the parabola opens downward.
One modelling sentence: a real situation governed by a "between two limits" condition, written as an inequality, with a note on what the solution range means in context.

Seed of Change: Solve linear and quadratic inequalities, including representing the solution set on a number line.

How this is assessed — the Haven Maths Rubric:

Strand	What we're looking for here
Conceptual Understanding	You can explain why a quadratic inequality's solution is sometimes an interval and sometimes two regions, using the graph.
Fluency & Accuracy	Linear and quadratic inequalities are solved correctly, including the sign flip and accurate open/closed boundaries.
Application to Problems	You can translate a real "within limits" situation into an inequality and interpret the solution range.
Independence & Reflection	The field guide is your own, and your explanation shows you reasoning from the picture rather than a memorised rule.

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