Demystifying - The inequality symbols

Introduction

Before this page: be able to put numbers in order, decimals and negatives included. Every symbol below is a statement about which of two quantities is the larger.

An equals sign settles one question: whether two quantities are the same. Most of the useful statements about numbers are not of that kind. A speed limit, a pass mark, a tolerance on a machined part, the range a temperature must stay inside — each of these describes how two quantities differ, or how far apart they are allowed to be, and none of them can be written with an equals sign.

Inequality symbols are the notation for those statements. Mathematics has a great many of them; six carry almost all of GCSE.

The first says that two values match. Each of the other five says how they differ, and every one is read the same way: the value on the left is described in terms of the value on the right.

Equal and not equal

0.6 = 0.60

Equal to

The two decimals are written differently, but both have the same value.

6 ≠ −6

Not equal to

Six and negative six are different values, so the equals sign is crossed out.

These two symbols settle only whether two values match, never which of them is the larger.

Less than and greater than

When two values are different, < and > tell us which is smaller. The point faces the smaller value, while the wide opening faces the larger value.

Interactive comparison number line A number line from zero to ten, with draggable points beginning at four and nine. Each point moves in increments of one tenth, and the value cards and comparison update as either point moves. 4 < 9 4 9 01 23 45 67 89 10

Drag either point along the line. The comparison can be read in either direction: for example, 4 < 9 means “4 is less than 9”, and also “9 is greater than 4”.

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Change the values

Decimals

0.72 < 0.8

Write 0.8 as 0.80. Both whole-number parts are zero, then 7 tenths is less than 8 tenths.

Negative numbers

−3 > −8

−3 is closer to zero, so it lies further right on the number line and is the larger value.

Adding “or equal to”

A short line underneath < or > adds “or equal to”. This means that the value at the limit is now allowed.

n ≤ 12

n is at most 12

This includes 12 and every value below it.

101112
t ≥ −4

t is at least −4

This includes −4 and every value above it.

−4−3−2

Other phrases with the same meanings are:

  • no more than 12 or a maximum of 12 means n ≤ 12

  • no less than 4 or a minimum of 4 means n ≥ 4

Values between two boundaries

Two inequalities can be joined to show that a value lies between two boundaries.

List the integers that satisfy −2 < n ≤ 3

Interactive integer boundary line A number line from negative four to five with a draggable open left boundary and closed right boundary. Each boundary moves in tenths, while only included integers are marked. −4−3 −2−1 01 23 45 −2 excluded 3 included 5 integer values in the range

  1. The open left boundary is not included.

  2. The closed right boundary is included.

  3. Read the integer points between the boundaries.

n = −1, 0, 1, 2 or 3

Common mistakes

  • Reversing < and >When the values differ, the point faces the smaller value and the wide opening faces the larger value.
  • Forgetting negative order−2 > −7 because −2 lies closer to zero.
  • Missing the equality linex ≤ 5 includes 5, but x < 5 does not.
  • Reading only left to right3 < n means that n is greater than 3.

Practice and continue

Practice

The inequality symbols

The less-than, greater-than, at-most, at-least and not-equal signs.

 Next lesson

Multiplying and dividing by powers of ten

The effect on every digit of multiplying or dividing by 10, 100, 0.1 and 0.01.