Inequalities Starters
Everything you need to know 'bout inequalities!
What is an Inequality?
An inequality is a statement that shows the relationship between two (or more) expressions with one of the following five signs: <, ≤, >, ≥, ≠.
x < y means "x is less than y"
x ≤ y means "x is less than or equal to y"
x > y means "x is greater than y"
x ≥ y means "x is greater than or equal to y"
x ≠ y means "x is not equal to y"
Like an equation, an inequality can be true or false.
34 − 12 > 5 + 2 is a true statement.
1 + 3 < 6 − 2 is a false statement.
1 + 3 ≤ 6 − 2 is a true statement.
1 + 3 ≠ 6 − 2 is a false statement.
−20 < −18 is a true statement.
Properties of inequalities
Let a, b and c be real numbers.
Firstly, the notation a < b means that a is smaller than b,
and the notation a > b means that a is greater than b.
Secondly, a ≤ b means that a is less than or equal to b,
and a ≥ b means that a is less than or equal to b.
1. Trichotomy Property
The following are true:
a < b
a = b
a > b
2. Transitive Property
If a < b and b < c then a < c
3. Additional Property
If a < b then a + c < b + c
4. Subtraction Property
If a < b then a - c < b - c
5. Multiplication Property
I. If a < b and c is positive then c X a < c X b
II. If a < b and c is negative c X a > c X b
Note:
1.If each inequality sign is reversed in the above properties, we obtain similar properties.
2.If the inequality sign < is replaced by <= (less than or equal) or the sign > is replaced by >= (greater than or equal), we also obtain similar properties.
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-3
< 5 - 2x
-8 < -2x
4 > x
(-inf, 4)
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AND
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5
- 2x < 9
-2x < 4
x > -2
(-2, inf)
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