An inequality is a mathematical statement that indicates the relationship between two quantities that are not equal. Inequalities can take various forms and are used to compare values, often to express that one value is greater than, less than, greater than or equal to, or less than or equal to another value. Here are some common types of inequalities:
1. Less than (`<`): One quantity is smaller than another.
Example: ( x < 5 ) means ( x ) is less than 5.
2. Greater than (`>`): One quantity is larger than another.
Example: ( y > 3 ) means ( y ) is greater than 3.
3. Less than or equal to (`≤`): One quantity is either less than or equal to another.
Example: ( z ≤ 10 ) means ( z ) is less than or equal to 10.
4. Greater than or equal to (`≥`): One quantity is either greater than or equal to another.
Example: ( w ≥ 0 ) means ( w ) is greater than or equal to 0.
5. Not equal to (`≠`): One quantity is not equal to another.
Example: ( a ≠ b ) means ( a ) is not equal to ( b ).
Inequalities are also used in algebraic expressions and can be solved to find the values of the variables that satisfy the condition. For example, the inequality ( 2x + 3 < 7 ) can be solved to find the values of ( x ) that make the inequality true.
Inequalities can also be linear, quadratic, or more complex, and they can be represented graphically on a number line or on a Cartesian plane, which can be very helpful for solving and understanding them.