Definitions [4]
Definition: Inequation
An inequation is a relation showing inequality between two quantities.
Symbols used:
-
> greater than
-
< less than
-
≥ greater than or equal to
-
≤ less than or equal to
Definition: Linear Inequation in One Variable
An inequation involving one variable of degree 1 is called a linear inequation in one variable.
General forms:
-
ax + b > c
-
ax + b < c
-
ax + b
-
ax + b
(where a, b, c are real numbers and a ≠ 0)
Definition: Replacement Set / Domain
The set from which values of the variable are taken is called the
replacement set or domain.
Definition: Solution Set
The set of all values from the replacement set that satisfy the inequality is called the solution set.
Key Points
Key Points: Properties of Inequation
| Operation on Both Sides | Inequality Sign | Example |
|---|---|---|
| Add the same number | No change | (x - 2 < 4 ⇒ x < 6) |
| Subtract the same number | No change | (x + 3 > 7 ⇒ x > 4) |
| × or ÷ by a positive number | No change | (x < 6 ⇒ 3x < 18) |
| × or ÷ by a negative number | Reverses | (-2x > 6 ⇒ x < -3) |
Key Points: Rules for Solving Linear Inequations
| Rule | Action | Effect |
|---|---|---|
| 1 | Transpose positive term to other side | Becomes − |
| 2 | Transpose negative term to other side | Becomes + |
| 3 | × / ÷ by +ve | Sign same |
| 4 | × / ÷ by −ve | Sign reverses |
| 5 | Change sign of all terms (× −1) | Sign reverses |
| 6 | Take reciprocals (both + or both −) | Sign reverses |
