Linear and quadratic inequalities

Q: How do you represent inequality boundaries using circles on a number line?

Use an open circle $(\circ)$ for strict inequalities () and a closed circle $(\bullet)$ for inclusive inequalities ($ \le $ or $ \ge $).

Q: When graphing inequalities in 2D space, how do you decide between drawing a solid or dashed line?

Use a dashed line for strict inequalities () and a solid line for inclusive inequalities ($ \le $ or $ \ge $).

Q: How can you determine which side of a boundary line to shade for a linear inequality?

Pick a test point (like $(0,0)$), substitute it into the inequality, and if it makes the statement true, shade the region containing that point.

Q: How do you determine the solution set from a sketch of a quadratic curve?

For $ > 0 $, choose the regions above the x-axis (the ends); for \( below the x-axis (the middle).

Q: What is set notation in the context of inequalities?

Set notation uses symbols like $\{x : x > 4\}$ to describe the collection of all values that satisfy an inequality.

Mathematics (Specification B) · Pearson Edexcel IGCSE · 5 min read · Updated: 7 Sept 2026

Welcome to Linear and Quadratic Inequalities

In your previous studies, you have spent a lot of time finding exact answers, like $x = 5$. But in the real world, things aren't always so exact. If you have $\$20$ to spend on lunch, you don't have to spend exactly $\$20$; you can spend any amount less than or equal to $\$20$. This is the heart of inequalities!

In this chapter, we will learn how to solve inequalities where the unknown is linear (like $x$) or quadratic (like $x^2$), and how to show those answers on number lines and graphs. Don't worry if this seems tricky at first—once you master a few simple rules, it becomes very logical!

1. The Basics: Inequality Symbols

Before we solve anything, we need to speak the language. Here are the symbols you must know:

$<$ : Less than (e.g., $3 < 5$)
$>$ : Greater than (e.g., $10 > 2$)
$\le$ : Less than or equal to
$\ge$ : Greater than or equal to

Quick Tip: Think of the symbol as an alligator's mouth. The alligator is hungry, so it always opens its mouth toward the bigger number!

2. Solving Linear Inequalities

Solving a linear inequality is almost exactly like solving a linear equation (which you can review in the Linear, quadratic and cubic equations chapter). Your goal is to get $x$ by itself.

The Golden Rule

There is one major difference between equations and inequalities: If you multiply or divide both sides by a negative number, you must flip the inequality sign!

Example 1: Solve $3x - 4 < 11$
1. Add $4$ to both sides: $3x < 15$
2. Divide by $3$: $x < 5$

Example 2: Solve $-2x + 6 \le 12$
1. Subtract $6$ from both sides: $-2x \le 6$
2. Divide by $-2$. Stop! We are dividing by a negative, so flip the sign: $x \ge -3$

3. Representing Solutions on a Number Line

Since an inequality represents a range of numbers, we often draw it on a number line to make it clear.

Open Circle $(\circ)$: Use this for $<$ or $>$. It means the number itself is not included.
Closed (Solid) Circle $(\bullet)$: Use this for $\le$ or $\ge$. It means the number is included.
The Line: Draw an arrow in the direction of the solution.

Example: To represent $x > 2$, you would draw an open circle at $2$ and a line pointing to the right.

Key Takeaway:

Treat inequalities like equations, but always remember to flip the sign when multiplying or dividing by a negative number!

4. Inequalities in Two-Dimensional Space

Sometimes you will be asked to represent inequalities on a coordinate grid ($x$ and $y$ axes). This is useful for "simultaneous linear inequalities"—where more than one rule applies at the same time.

Step-by-Step: Drawing a Region

1. Draw the boundary line: Treat the inequality as an equation (e.g., for $y > x + 2$, draw the line $y = x + 2$).
2. Solid vs. Dashed: Use a dashed line for $<$ or $>$. Use a solid line for $\le$ or $\ge$.
3. Shade the region: Pick a test point (like $(0,0)$) and see if it makes the inequality true. If it does, shade the side containing that point. If not, shade the other side.

Did you know? In the exam, you might be asked to shade the region that satisfies several inequalities at once. The "solution" is the area where all the individual shaded regions overlap.

5. Quadratic Inequalities

Quadratic inequalities involve an $x^2$ term, such as $x^2 - 5x + 6 < 0$. These are solved in a slightly different way because the "direction" of the answer depends on the shape of the curve.

How to Solve Quadratic Inequalities

Step 1: Rearrange. Get everything onto one side so the other side is $0$ (e.g., $ax^2 + bx + c > 0$).

Step 2: Find the "Critical Values". Temporarily treat it as an equation $ax^2 + bx + c = 0$. Solve it by factorising or using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. These answers are your boundaries.

Step 3: Sketch the graph. Draw a simple $U$-shaped curve (a parabola) that crosses the x-axis at your critical values.

Step 4: Identify the region.

If the inequality is $> 0$, you want the parts of the curve above the x-axis (the "ends").
If the inequality is $< 0$, you want the part of the curve below the x-axis (the "middle" bit).

Example: Solve $x^2 - 9 \le 0$
1. Critical values: $x^2 - 9 = 0 \implies (x-3)(x+3) = 0$. So, $x = 3$ and $x = -3$.
2. Sketch the $U$-shape crossing at $-3$ and $3$.
3. Since we want $\le 0$, we look below the x-axis.
4. The solution is the interval between the two numbers: $-3 \le x \le 3$.

Representing Quadratic Solutions

Just like linear inequalities, these can be shown on a number line. For $-3 \le x \le 3$, you would draw solid circles at $-3$ and $3$ and connect them with a single line.

Common Mistake to Avoid:

Don't try to solve $x^2 > 9$ by just saying $x > 3$. You must remember that $(-4)^2$ is also greater than $9$! Always sketch the graph to find both parts of the solution: $x > 3$ and $x < -3$.

Quick Review

Linear: Solve like an equation; flip sign for negatives; represent with circles and arrows.
2D Space: Dashed lines for $<$ or $>$; solid for $\le$ or $\ge$; use test points for shading.
Quadratic: Find critical values; sketch the parabola; choose the region above or below the axis.
Notation: Be comfortable with "set notation" symbols like $\{x : x > 4\}$ which simply means "the set of all $x$ such that $x$ is greater than $4$".

Note: For more practice on solving the equations used to find critical values, check the chapter on Factorisation, factor theorem and algebraic division.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are the four primary inequality symbols and their meanings?

$ < $ is less than, $ > $ is greater than, $ \le $ is less than or equal to, and $ \ge $ is greater than or equal to.

What is the "Golden Rule" when multiplying or dividing an inequality by a negative number?

You must flip (reverse) the inequality sign whenever you multiply or divide both sides by a negative number.

How do you represent inequality boundaries using circles on a number line?

Use an open circle $(\circ)$ for strict inequalities ($ < $ or $ > $) and a closed circle $(\bullet)$ for inclusive inequalities ($ \le $ or $ \ge $).

When graphing inequalities in 2D space, how do you decide between drawing a solid or dashed line?

Use a dashed line for strict inequalities ($ < $ or $ > $) and a solid line for inclusive inequalities ($ \le $ or $ \ge $).

How can you determine which side of a boundary line to shade for a linear inequality?

Pick a test point (like $(0,0)$), substitute it into the inequality, and if it makes the statement true, shade the region containing that point.

What are "critical values" in the context of solving a quadratic inequality?

Critical values are the solutions to the quadratic equation $ax^2 + bx + c = 0$, representing where the curve crosses the x-axis.

How do you determine the solution set from a sketch of a quadratic curve?

For $ > 0 $, choose the regions above the x-axis (the ends); for $ < 0 $, choose the region below the x-axis (the middle).

What is set notation in the context of inequalities?

Set notation uses symbols like $\{x : x > 4\}$ to describe the collection of all values that satisfy an inequality.

Ready to test yourself?

Turn these notes into exam-style practice. Get unlimited AI questions on this topic with instant marking and explanations.

Practise This Topic

More Mathematics (Specification B) chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practise This Topic Now