FORM 2 • TERM 1 • LESSON 11

An equation says two things are equal. Very often in real life, though, we only know that something is more than, less than, at least or at most some amount. Statements like these are called inequalities. In this lesson you will learn what inequalities mean in everyday life, how to write them with mathematical symbols, and how to turn word problems into inequalities. In the next lesson you will solve them.

Learning outcomes

By the end of this lesson you should be able to:

• 2.6.4.1 – Explain what inequalities are, with examples from society, politics and economics.
• 2.6.4.2 – Use mathematical language and symbols to write inequalities.
• 2.6.4.3 – Rewrite word problems as inequalities.
• 2.6.4.4 – Identify the set of numbers that the solution of an inequality belongs to.

Inequalities in everyday life

In everyday speech, “inequality” often means that people are not treated the same or do not have the same amount – for example, the gap between high and low incomes. In mathematics, an inequality is a statement that one quantity is bigger or smaller than another, or that a quantity must stay above or below a limit. Limits like this are everywhere:

• Society: a road speed limit of 50 km/h means your speed s must satisfy s ≤ 50. A lift with a limit of 8 people needs p ≤ 8.
• Politics: in Trinidad and Tobago you must be at least 18 years old to vote, so a voter’s age a satisfies a ≥ 18. The House of Representatives has 41 elected seats, so a party needs at least 21 of them, s ≥ 21, to have a majority.
• Economics: a family budget of $600 a week for food means spending f ≤ 600. A business needs its income to be more than its costs to make a profit: income > costs.

The inequality symbols

Symbol	Meaning	Example	Read as
<	is less than	x < 5	x is less than 5
>	is greater than	x > −2	x is greater than −2
≤	is less than or equal to	x ≤ 10	x is at most 10
≥	is greater than or equal to	x ≥ 18	x is at least 18

The “open mouth” of < and > always faces the bigger quantity. The line under ≤ and ≥ means “or equal to”, so the end number itself is allowed. You can read an inequality either way round: x < 5 means the same as 5 > x.

Key phrases

Words	Symbol	Example
more than, greater than, over, above, exceeds	>	more than 30 students: n > 30
less than, fewer than, under, below	<	fewer than 12 eggs: e < 12
at least, no less than, a minimum of, not under	≥	at least 18 years old: a ≥ 18
at most, no more than, a maximum of, not over, up to	≤	at most 23 kg: m ≤ 23

A quantity that is between two values uses two signs at once. “The temperature was between 24 °C and 31 °C, inclusive” is 24 ≤ t ≤ 31.

Writing word problems as inequalities

1. Choose a letter for the unknown quantity and say what it stands for.
2. Write an expression for the quantity that is limited.
3. Find the key phrase and pick the correct symbol.
4. Write the inequality, with the limit on the other side.

Which numbers make sense?

The set of numbers a solution can belong to depends on what the letter stands for. Use the number families from Lesson 1:

The letter stands for…	Sensible set of numbers	Example
a count of people, tickets, buses	Whole numbers (0, 1, 2, …)	t ≤ 4 gives t = 0, 1, 2, 3 or 4
a count that must be at least one	Natural numbers (1, 2, 3, …)	1 ≤ b ≤ 3 gives b = 1, 2 or 3
a temperature in whole degrees, a floor in a building with basements	Integers (…, −2, −1, 0, 1, …)	−3 ≤ f ≤ 2
a length, mass, time or speed	Any number, including fractions and decimals	m ≤ 23 includes 22.5 kg
an amount of money	Decimals with two places (cents)	c ≤ 150 includes $149.99

Worked examples

Example 1: Reading inequalities

Question: Write in words:
(a) h > 140
(b) n ≤ 35
(c) −5 < t < 0.

1. (a) > means “greater than”.
2. (b) ≤ means “less than or equal to”, or “at most”.
3. (c) t is between −5 and 0, not including either end.

Answer:
(a) h is greater than 140
(b) n is at most 35
(c) t is between −5 and 0.

Example 2: Writing a limit

Question: A suitcase on a flight must have a mass of no more than 23 kg. Write an inequality for its mass m kg.

1. “No more than” means at most: ≤.

Answer: m ≤ 23.

Example 3: A ride with a height rule

Question: To go on a ride you must be at least 120 cm tall. Write an inequality for an allowed height h cm.

1. “At least” means greater than or equal to: ≥.

Answer: h ≥ 120.

Example 4: A two-operation word problem

Question: Concert tickets cost $35 each. Jenna has $150. Write an inequality for the number of tickets t she can buy, and say which set of numbers t belongs to.

1. The cost of t tickets is 35t dollars.
2. She cannot spend more than $150: 35t ≤ 150.
3. You cannot buy part of a ticket, so t is a whole number. (35 × 4 = 140 is fine; 35 × 5 = 175 is too much.)

Answer: 35t ≤ 150, where t is a whole number (so t = 0, 1, 2, 3 or 4).

Example 5: A number puzzle

Question: When a number is doubled and 3 is added, the result is greater than 17. Write an inequality.

1. Doubled and 3 added: 2n + 3.
2. “Greater than” is >.

Answer: 2n + 3 > 17.

Example 6: Making a profit

Question: A juice stall makes $4 profit on each cup. The owner wants a profit of at least $300 on Saturday. Write an inequality for the number of cups c, and say what kind of number c is.

1. Profit = 4c dollars.
2. “At least” gives ≥.
3. Cups are counted, so c is a whole number.

Answer: 4c ≥ 300, where c is a whole number.

Common mistakes

• Using < for “at most”. At most 8 people means 8 is allowed, so the sign is ≤.
• Turning the sign the wrong way. “At least 18” is a ≥ 18, not a ≤ 18.
• Mixing up “less than” order. “x is less than 5” is x < 5; “5 less than x” is the expression x − 5.
• Allowing impossible answers. If t counts tickets, t = 2.5 is not a real answer.
• Using an equals sign. A limit such as “at most 23 kg” is an inequality, not an equation.

Practice questions

Try these yourself first, then open each answer to check your working.

Quick check

Previous: Linear Equations: Letters on Both Sides, Brackets and Word Problems

Next: Solving Inequalities and Showing Answers on a Number Line and in Set Builder Notation

Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 2, Term 1, Strand 2.6 Algebra, Topic 2.6.4 Solution of Linear Inequalities – Learning Outcomes 2.6.4.1 (inequalities in society, politics and economics), 2.6.4.2 (mathematical language for inequalities), 2.6.4.3 (rewriting word problems as inequalities) and 2.6.4.4 (the set of numbers a solution belongs to).

Attributions to media used in this post:

50 km speed limit road sign 2.jpg – HutheMeow – CC BY-SA 4.0 (via Wikimedia Commons)

Updates

2026/10/23

– First version of this lesson.

2026/10/25

– Put each lettered part of the questions and answers on its own line.

© 2026 Vedesh Kungebeharry. All rights reserved.