Definitions [4]
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
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)
The set from which values of the variable are taken is called the
replacement set or domain.
The set of all values from the replacement set that satisfy the inequality is called the solution set.
Key Points
| 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) |
| 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 |
Endpoints:
-
< or > → hollow circle (endpoint not included)
-
≤ or ≥ → solid/dark circle (included)
Direction:
-
x > a: shade to the right
-
x < a: shade to the left
-
a < x ≤ b: between a and b, left open, right closed
| Operation | Meaning | Set Form |
|---|---|---|
| AND | Common values satisfying both | (P ∩ Q) |
| OR | Values satisfying either or both | (P ∪ Q) |
| P but not Q | Belong to (P), not to (Q) | (P - Q = P ∩ Q') |
| Q but not P | Belong to (Q), not to (P) | (Q - P = P' ∩ Q) |
-
If product < 0 ⇒ one +, one − ⇒ solution between roots
a < x < b (or b < x < a) -
If product > 0 ⇒ both + or both − ⇒ solution outside roots
x < a or x > b
