Inequalities in one Variable
In this section we will learn how to use the relation symbols in word sentences and
how to translate them into mathematical sentences.
Comparison Symbols
| English terms; |
Relation Symbols and their |
Logical (negative) equivalents:
|
| English |
Relation |
Negation (opposite)
|
| equals |
= is, equals |
≠ not equal (not an equivalent) |
| less than |
< is less than |
 is not greater than or equal
|
| at most |
≤ is less than or equal |
is not greater than |
| beyond |
> is greater than |
 is not less than or equal |
| at least |
≥ is greater than or equal |
is not less than |
Rules of Order of Operations
Please Pardon My Dear Aunt Sally
Parentheses (Grouping): Work from inside out.
Powers (Exponents): Simplify all powers first.
Multiply and/or
Divide: Work left to right (watch signs).
Add and/or Subtract: Combine like terms only.
Properties of Inequalities
For any real numbers a, b, and c:
1. EQUIVALENT PROPERTY:
a < b ↔ b > a
2. ADDITION PROPERTY:
If a < b is true then a + c < b + c is true.
3. MULTIPLICATION PROPERTY:
For c > 0, if a < b is true then a ·c < b ·c is true.
For c < 0, if a < b is true then a ·c > b ·c is true.
NOTE: Negation reverses every sign in its path. (-1)·(-2 < 3)
↔ +2 > -3
