Inequalities

About Inequalities

An inequality question gives a few relations between letters and asks which conclusion must be true. The words must be taken strictly: an option is correct only when it follows from the statements in every case, not merely in one arrangement you can picture. Joining the statements into a single chain is what makes that check possible.

What you need to understand

Each statement is a link: the letter on the wide side of the symbol is larger.

Links that share a letter can be joined, giving a longer chain in one direction.

Once the letters form a single chain, any two of them can be compared by their positions along it.

A conclusion follows only if it is true in every arrangement the statements allow, not just some.

Reversing a link produces a statement the chain does not support, which is the most common wrong option.

How to work through these questions

Rewrite every statement so the smaller letter is on the left, whichever way it was written.

Join the links into one chain, working outwards from a letter that appears twice.

Read off the position of each letter in the chain.

Test each option by asking whether the chain forces it, rather than whether the option is plausible.

Reject any option that reverses a link or that needs information the statements never gave.

Mistakes that cost marks

Accepting a conclusion that is possible rather than certain.
Reading a statement the wrong way round, since the wider end of the symbol is the larger value.
Assuming two letters are ordered when no chain connects them.
Confusing the weak and strict forms: knowing that a value is at least something does not make it strictly greater.
Joining two links that do not actually share a letter.

Worked example

Statements: A < B, B < C, C < D. Which of the following is definitely true?

All three statements already run in the same direction, so the letters form one chain: A, then B, then C, then D.
In a chain written from smallest to largest, any letter is smaller than every letter to its right.
Testing the option A < D: A is at the start of the chain and D at the end, so A is certainly smaller.
The chain forces the relation, so the option follows in every arrangement.

Answer: A < D

Practice questions with answers

A few Inequalities questions with the full solution shown, so you can see how the method is applied before you attempt the timed set.

Question 1

Statements: E > C, G > B, C > G. Which of the following is definitely true?

A E < C
B C > E
C G < E
D E < G

Answer: Option C — with explanation

The statements give one chain: B < G < C < E. A 'less than' relation holds between any two members of the chain when the smaller is written on the left, so G < E follows. The other options reverse a link, which the statements do not allow. Common mistakes - reverses a link, so it does not follow: C > E - reverses a link, so it does not follow: E < C - reverses a link, so it does not follow: E < G

Question 2

Statements: C > A, F < H, H < A. Which of the following is definitely true?

A F > C
B C < H
C F > A
D H < C

Answer: Option D — with explanation

The statements give one chain: F < H < A < C. A 'less than' relation holds between any two members of the chain when the smaller is written on the left, so H < C follows. The other options reverse a link, which the statements do not allow. Common mistakes - reverses a link, so it does not follow: F > A - reverses a link, so it does not follow: C < H - reverses a link, so it does not follow: F > C

Question 3

Statements: C > B, A < B, H < A. Which of the following is definitely true?

A A < C
B A < H
C A > C
D C < B

Answer: Option A — with explanation

The statements give one chain: H < A < B < C. A 'less than' relation holds between any two members of the chain when the smaller is written on the left, so A < C follows. The other options reverse a link, which the statements do not allow. Common mistakes - reverses a link, so it does not follow: C < B - reverses a link, so it does not follow: A < H - reverses a link, so it does not follow: A > C

Question 4

Statements: C < D, H > B, H < C. Which of the following is definitely true?

A B < C
B B > C
C D < B
D D < H

Answer: Option A — with explanation

The statements give one chain: B < H < C < D. A 'less than' relation holds between any two members of the chain when the smaller is written on the left, so B < C follows. The other options reverse a link, which the statements do not allow. Common mistakes - reverses a link, so it does not follow: D < B - reverses a link, so it does not follow: D < H - reverses a link, so it does not follow: B > C

Question 5

Statements: F < C, C < H, G > H. Which of the following is definitely true?

A F < H
B F > C
C F > H
D H > G

Answer: Option A — with explanation

The statements give one chain: F < C < H < G. A 'less than' relation holds between any two members of the chain when the smaller is written on the left, so F < H follows. The other options reverse a link, which the statements do not allow. Common mistakes - reverses a link, so it does not follow: F > H - reverses a link, so it does not follow: H > G - reverses a link, so it does not follow: F > C

Frequently asked questions

What does definitely true mean here?

It means the relation holds in every arrangement of values that satisfies the statements. A relation that is merely possible, or true in one arrangement you happen to choose, does not count.

How do I join the statements?

Look for a letter that appears in two of them and use it to link them together. Joining the shared letters gives one chain, and the chain is what you read the answer from.

Why is reversing a statement wrong?

Because the statements only tell you about one direction. If A is smaller than B, that says nothing about whether B is smaller than some other letter unless a further statement says so.

What is the difference between the weak and strict symbols?

The weak form allows the two values to be equal, while the strict form does not. Tightening a weak relation into a strict one is a common trap, so treat them carefully.

Do I need to test each option?

Yes, and quickly. Ask of each one whether the chain forces it. The option the chain forces is the answer, and everything else is either possible or false.