Inequalities
Statements that express the inequality of algebraic expressions are called
inequalities. The symbols that we use to express inequality are given below with
their meanings.
Inequality Symbols
Symbol 
Meaning 
< 
Is less than 
≤ 
Is less than or equal to 
> 
Is greater than 
≥ 
Is greater than or equal to 
It is clear that 5 is less than 10, but how do we compare 5 and 10? If we
think of negative numbers as debts, we would say that 10 is the larger debt.
However, in algebra the size of a number is determined only by its position on
the number line. Fow two numbers a and b we say that a is less than b if and
only if a is to the left of b on the number line. To compare 5 and 10, we
locate each point on the number line (see figure below). Because 10 is to the
left of 5 on the number line, we say that 10 is less than 5. In symbols,
10 < 5.
We say that a is greater than b if and only if a is to the right of b on the
number line. Thus we can also write
5 > 10.
The statement a ≤ b is true if a is less than b
or if a is equal to b. The statement a ≥ b is true if a is greater than b or if
a equals b. For example, the statement 3 ≤ 5 is true, and so is the statement 5
≤ 5.
Example
Inequalities
Determine whether each statement is true or false.
a) 5 < 3
b) 9 > 6
c) 3 ≤ 2
d) 4 ≥ 4
Solution
a) The statement 5 < 3 is true because 5 is to
the left of 3 on the number line. In fact, any negative number is less than any
positive number.
b) The statement 9 > 6 is false because 9
lies to the left of 6.
c) The statement 3 ≤ 2 is true because 3 is
less than 2.
d) The statement 4 ≥ 4 is true because 4 = 4 is
true.
