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.
|
