Inequalities

Learn it by playing

Answer these questions to earn energy, then fish and explore. No account needed.

For teachers: ready-to-use lesson slides, revision notes, diagrams for Inequalities (Maths [CIE], Core) — use them in your lesson, or run the topic as a live class game.

Notes

Interpreting Inequalities

An **inequality** compares two values: **>** (greater than), **<** (less than), **≥** (greater than or equal to), **≤** (less than or equal to).
**Strict inequalities** $use > or < and$ do **not** include the endpoint.
**Non‑strict inequalities** $use \ge or \le and$ **do** include the endpoint.
Example: $x > 5$ means x can be 6, 7, 8, … (not 5).
Example: $x \le 10$ means x can be 10, 9, 8, … (includes 10).

Finding Integer Solutions

When asked to list **integer** values satisfying an inequality, check whether each endpoint is included.
For $3 \le x \le 6$ , integers are 3, 4, 5, 6 (both endpoints included).
For $3 \le x < 6$ , integers are 3, 4, 5 (6 not included).
For $3 < x \le 6$ , integers are 4, 5, 6 (3 not included).
For $3 < x < 6$ , integers are 4, 5 (neither endpoint included).
Remember that **zero** and **negative whole numbers** are integers unless stated otherwise.
To satisfy two inequalities, list integers for each and find the **intersection** (common values).

Representing Inequalities on a Number Line

Use a **closed circle** (●) for endpoints that are included $(\le or \ge )$ .
Use an **open circle** (○) for endpoints that are not included $(< or >)$ .
Connect circles with a **solid line** between them for a range.
For a one‑sided inequality $(e.g., x > 5)$ , draw an **arrow** from the circle pointing in the direction of the inequality.
Example: $-2 \le x < 1$ has a closed circle at −2, open circle at 1, and a line between them.
Example: $t < 3$ has an open circle at 3 and an arrow to the left.

Solving Simple Inequalities

Inequalities can be solved similarly to equations, but the direction of the inequality sign must be preserved.
Example: Solve $y - 2 > 5$ → add 2 to both sides → $y > 7$ .
The solution $y > 7$ means y can be any number greater than 7 (not including 7).
For inequalities involving squares, consider both positive and negative roots $(e.g., m^{2} > 9$ → $m < -3$ or $m > 3)$ .

Finding the Largest or Smallest Integer

The **smallest integer** satisfying $x > 6.5$ is 7 (since 6 is not greater than 6.5).
The **largest odd integer** satisfying $y \le 6.5$ is $5 (since 7 > 6.5$ and 5 is odd).
Always check whether the endpoint is included when determining the extreme integer.

Number line showing −2 ≤ x < 1

Number line showing t < 3

Number line showing x > 5

Number line showing 3 ≤ x ≤ 6

Practice questions

Free preview — 8 of 40 questions. Sign up to see them all.

1.What does the inequality symbol ≤ mean?
Easy
- Aless than or equal to
- Bgreater than or equal to
- Cless than
- Dgreater than
2.Which of the following is a strict inequality?
Easy
- A $x > 5$
- B $x \ge 5$
- C $x \le 5$
- D $x = 5$
3.n is an integer. $-1 \le n < 4$ . List the possible values of n.
Easy
- A−1, 0, 1, 2, 3
- B−1, 0, 1, 2, 3, 4
- C0, 1, 2, 3
- D−1, 0, 1, 2
4.Write down the inequality shown on the number line: an open circle at 3 and an arrow pointing to the right.
Easy
- A $x > 3$
- B $x \ge 3$
- C $x < 3$
- D $x \le 3$
5.m is an integer such that $-2 < m \le 3$ . Write down all the possible values of m.
Easy
- A−1, 0, 1, 2, 3
- B−2, −1, 0, 1, 2, 3
- C−1, 0, 1, 2
- D0, 1, 2, 3
6.Show the inequality $x < 3$ on a number line. Which number line is correct?
Medium
- AOpen circle at 3, arrow left
- BClosed circle at 3, arrow left
- COpen circle at 3, arrow right
- DClosed circle at 3, arrow right
7.n is an integer. Write down all the values of n such that –2 $\le n < 3$ .
Medium
- A–2, –1, 0, 1, 2
- B–2, –1, 0, 1, 2, 3
- C–1, 0, 1, 2
- D–2, –1, 0, 1
8.On the number line, represent the inequality $y \le 1$ . Which representation is correct?
Medium
- AClosed circle at 1, arrow left
- BOpen circle at 1, arrow left
- CClosed circle at 1, arrow right
- DOpen circle at 1, arrow right

Summary

## Inequalities ### Interpreting Inequalities - **Inequality**: compares values using >, <, ≥, ≤. - **Strict inequalities**: > and < (endpoint not included). - **Non-strict**: ≥ and ≤ (endpoint included). #### Finding integer values - List whole numbers between endpoints, checking inclusion. - Example: −4 ≤ x < 2 → x = −4, −3, −2, −1, 0, 1. - For two inequalities, list integers satisfying both. - If no integer specified, do not assume integer. ### Representing on a Number Line - **Open circle** (○) for < or > (not included). - **Closed circle** (●) for ≤ or ≥ (included). - Draw a line connecting circles, or an arrow for one endpoint. - Example: −2 ≤ x < 1 → closed at −2, open at 1, line between. - Example: t < 3 → open at 3, arrow left. ### Key Points - Integers include zero and negatives. - For smallest/largest integer: e.g., smallest integer > 6.5 is 7.

Key terms

Inequality: A statement that one value is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another.
Strict inequality: An inequality using > or <; the endpoint is not included.
Integer: A whole number (positive, negative, or zero).
Number line: A visual representation of numbers; used to show inequalities with open/closed circles and lines or arrows.
Open circle: A circle drawn above a number on a number line to indicate that the endpoint is not included (used with < or >).
Closed circle: A filled circle drawn above a number on a number line to indicate that the endpoint is included (used with ≤ or ≥).

Slides

What is an Inequality?

An inequality tells you that something is greater than (>) or less than (<) something else.
For example, $x > 5$ means x is greater than 5.
Inequalities can also include $'equal to': \ge$ means greater than or equal $to, \le$ means less than or equal to.
Strict inequalities $(< or >)$ do not include the end point; $non-strict (\le or \ge )$ do include it.

Which of the following is true about $x > 5$ ?

x can be 5
x can be 6
x can be 4
x can be 5.5

Finding Integers that Satisfy Inequalities

Integers are whole numbers (including zero and negatives).
To list integers satisfying an inequality, check whether each end point is included.
For example, $3 \le x \le 6$ gives $x = 3, 4, 5, 6 (both$ ends included).
$3 \le x < 6$ gives $x = 3, 4, 5 (6$ not included).
If only one end point is given, there are infinitely many integers $(e.g., x > 2$ gives 3, 4, 5, ...).

List all the integer values of x that satisfy $-4 \le x < 2: x = ____, ____, ____, ____, ____, ____$

Representing Inequalities on a Number Line

Use circles above the end points: open circle $for < or > (not included)$ , closed circle $for \le or \ge (included)$ .
Connect the circles with a horizontal line.
For inequalities with only one end point, draw an arrow to the left $(for < or \le )$ or right $(for > or \ge )$ .

Which of these are true about representing inequalities on a number line?

An open circle means the end point is included.
A closed circle means the end point is included.
The inequality $x > 3$ is shown with an open circle at 3 and an arrow to the right.
The inequality $x \le 2$ is shown with a closed circle at 2 and an arrow to the left.

Example: Representing -2 ≤ x < 1

-2 is included, so use a closed circle at -2.
1 is not included, so use an open circle at 1.
Draw a line connecting the circles.

How would you represent $x < 3$ on a number line?

Open circle at 3, arrow to the right
Closed circle at 3, arrow to the left
Open circle at 3, arrow to the left
Closed circle at 3, arrow to the right

Finding Integers from Two Inequalities

List integers for each inequality separately.
Find the common integers that satisfy both.
For example, $0 < x < 5$ gives $x = 1, 2, 3$ , 4; $x \ge 3$ gives $x = 3, 4, 5, 6$ , ...; common: $x = 3, 4$ .

List all the integers that satisfy both $x \le 8$ and $x > 3: x = ____, ____, ____, ____, ____$

Smallest or Largest Integer

You may be asked for the smallest or largest integer satisfying an inequality.
For $x > 6.5$ , the smallest integer is 7 (since 6 is $not > 6.5)$ .
For $y \le 6.5$ , the largest integer is 6 (since 7 is $not \le 6.5)$ .

What is the largest integer that satisfies $y \le 6.5$ ?

Reading Inequalities from a Number Line

To write the inequality shown on a number line, identify the end points and whether they are included.
Open circle means $< or$ >; closed circle means $\le or \ge$ .
If there is an arrow, the inequality extends infinitely in that direction.

Which of these correctly describe the inequality shown by a number line with an open circle at -4 and a closed circle at 5?

The inequality $is -4 < x \le 5$
The inequality $is -4 \le x < 5$
The inequality $is -4 < x < 5$
The inequality $is -4 \le x \le 5$

Worked Example: List Integers

List all integer values of x that satisfy $-4 \le x < 2$ .
Start at -4 (included) and go up to 1 (since 2 is not included).
Answer: $x = -4, -3, -2, -1, 0, 1$ .

List all the integer values of y that satisfy $-5 < y \le 0: y = ____, ____, ____, ____, ____$

Summary

Inequalities compare values using <, >, ≤, ≥.
Integers are whole numbers; list them by checking end points.
On a number line, use open circles $for < or >$ , closed $for \le or \ge$ .
To satisfy multiple inequalities, find common integers.

Which inequality is shown by a number line with a closed circle at -2 and an open circle at 3?

$-2 \le x < 3$
$-2 < x \le 3$
$-2 < x < 3$
$-2 \le x \le 3$

Flashcards

What does the inequality $x > 5$ mean?

x is greater than 5. x could be 6, 7, 8, 9, ...

What does the symbol ⩾ mean?

Greater than or equal to.

What does the symbol ⩽ mean?

Less than or equal to.

What are strict inequalities?

Inequalities that do not include being equal, $i.e., > and <$ .

List the integer values of n that satisfy −1 ⩽ $n < 4$ .

$n = -1, 0, 1, 2, 3$

List the integer values of y that satisfy $-5 < y$ ⩽ 0.

$y = -4, -3, -2, -1, 0$

List the integer values of m that satisfy $-2 < m$ ⩽ 3.

$m = -1, 0, 1, 2, 3$

How do you represent an inequality on a number line?

Draw circles above the end points: open circle $for < or >$ , solid circle for ⩽ or ⩾. Connect them with a horizontal line, or use an arrow for one end point.

Show the inequality $x < 3$ on a number line.

An open circle at 3 with a horizontal arrow pointing to the left.

Represent the inequality y ⩽ 1 on a number line.

A solid circle at 1 with a horizontal arrow pointing to the left.

Which inequality is shown by a diagram with a solid circle at -4 and an open circle at 5, connected by a line?

−4 ⩽ $x < 5$

Show the inequality −2 ⩽ $y < 1$ on a number line.

Solid circle at -2, open circle at 1, with a line connecting them.

List all integer values of n that satisfy $-3.4 < n$ ⩽ 2.

$n = -3, -2, -1, 0, 1, 2$

List all integers that satisfy $x \le 8, x > 2$ , and $x > 3$ .

$x = 4, 5, 6, 7, 8$

What is the largest odd integer that satisfies $y \le 6.5$ ?

Write down the inequality shown in a diagram with an open circle at -2 and a solid circle at 3, connected by a line.

$-2 < n$ ⩽ 3

Solve $y - 2 > 5$ .

$y > 7$

Represent the inequality $-2 < n$ ⩽ 3 on a number line.

Open circle at -2, solid circle at 3, with a line connecting them.

Given $m^{2} > 9$ , circle the possible value of m from: 2, 7, 8, 2.8, 3, −7, 2.

−7

List all integers that satisfy $x \le 7, 2 < x \le 8$ , and $4 \le x \le 8.5$ .

$x = 4, 5, 6, 7$

What is the smallest integer that satisfies $x > 6.5$ ?

List the integer values of x that satisfy $-4 \le x < 2$ .

$x = -4, -3, -2, -1, 0, 1$

Represent the inequality $t < 3$ on a number line.

Open circle at 3 with a horizontal arrow pointing to the left.

Unlock all 40 questions, slides, flashcards & more

Create a free account to see every question, the slides, flashcards and revision notes for this topic.

Past papers

Past-paper practice for this topic is coming soon.

🗂️ Coming soon