The inequality symbols

Before this page: be able to put numbers in order, decimals and negatives included.

Equal and not equal

An equals sign tells us that two values are the same. To show that they are different, we use ≠. Other comparison symbols tell us which value is larger, or where a boundary lies.

  • <less than
  • =equal to
  • >greater than
  • ≤less than or equal to
  • ≠not equal to
  • ≥greater than or equal to

Every one is read left to right: how the value on the left compares with the value on the right.

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.

= and ≠ say only whether two values match, not which is the larger.

Less than and greater than

When two values are different, < and > show 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

4 < 9 and 9 > 4 record the same comparison: 4 is less than 9, and 9 is greater than 4.

<

Decimals

0.72 < 0.8

Write 0.8 as 0.80. Both whole-number parts are zero, and 72 hundredths is less than 80 hundredths, as 7 tenths is less than 8.

Negative numbers

−3 > −8

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

Including a boundary

A short line underneath < or > adds “or equal to”, so the value at the boundary is allowed.

n ≤ 12

n is at most 12

n ≤ 12 includes 12 and every value below it.

101112
t ≥ −4

t is at least −4

t ≥ −4 includes −4 and every value above it.

−4−3−2

For a speed limit of 30 miles per hour, the speed in miles per hour must satisfy speed ≤ 30. The limit itself is allowed.

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

Between two boundaries

−2 < n ≤ 3 joins two inequalities: n is greater than −2 and at most 3.

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 left symbol, <, has no line under it, so the left boundary is not included.

  2. The right symbol, ≤, has the line, so the right boundary is included.

  3. The integers between them are the solutions.

n = −1, 0, 1, 2 or 3

That list answers “write down all the integer values of n that satisfy −2 < n ≤ 3”, the form the question takes on a paper.

Key points

  • The pointed end of < or > faces the smaller value.
  • The symbols ≤ and ≥ allow equality; < and > do not.
  • Both parts of a double inequality must hold. Check each boundary separately.

Continue

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.
Back to Written methods