Inequalities


The notation

x > 2

is shorthand for

'x is greater than 2'.

This could also be written as

2 < x

The actual order that these inequalities are written down is not mathematically important - what is important is to kep the same numbers and symbols in their right position relative to the inequality signs. Although obviously, in practice, a certain way of presenting the inequality might be more 'elegant' than other(s).

A variation on this would be

x ≥ 2

which is shorthand for

'x is greater than or equal to 2'.


The number lines above are sometimes used to visually represent the range of values indicated by an inequality. These two displayed are intended to show the subtle difference between the two inequalties mentioned at the beginning. The first inequality (x>2) does not actrually include '2' in its range - the white circle indicates this. The second inequality (x≥2) has an almost identical range except that it does include '2' in its range. The black circle indicates that '2' is indeed included within the range.


If x represents a value bracketed between two boundaries, say 2 and 5, then that could be represented as

2 < x < 5

where x is greater than 2 but is less than 5.

A slight variation on this could be

2 < x ≤ 5

where this time x is still greater than 2, but x is not just less than 5 - now it could equal 5 as well.

Quick Quiz     Write these statements using mathematical symbols  

  a)   x is less than -1  

  b)  y is greater than or equal to seventeen

  c)   x is less than or equal to ten and greater than five

  d)  3x is greater than one and less than one hundred

  e)  x + y is less than five and greater than or equal to minus seven.

Manipulation of Inequalities


Inequalities can be processed just like equations when adding or subtracting values to both sides.

Multiplying and dividing also follows the same procedure just as long as the multiplier or divisor are positive. If they are negative, the inequality sign would need to be reversed. Any unknown algrbraic value, e.g x, cannot therefore be used as a multiplier or divisor by virtue of the fact that we do not know whether it is a positive or negative value.