Inequalities

In the topic inequalities, we generally come across various sets on real line. So, let us first know about these sets which are generally called intervals.

Closed Interval
Let a and b be two given real numbers such that a < b. Then the set of all real numbers x such that a≤x≤b$a\le x\le b$ is a closed interval and is denoted by [a, b].
[a, b] = {x∈R;a≤x≤b}$\left\{x\in R;a\le x\le b\right\}$
 [a, b] is the set of all real numbers lying between a and b including the end points.

Open interval
Let a and b two given real numbers such that a < b. Then the set of all real numbers x such that a < x < b is a open interval and is devoted by (a, b).
[a, b] = {x R: a < x < b}
(a, b) is the set of all real numbers lying between and b excluding the end points.

Semi open or Semi closed interval
Let a and b be two real numbers such that a < b. Then the sets (a, b] = {x R: a < x ≤$\le$  b} and [a, b) = {x R: a ≤$\le$  x < b} are known as semi-open or semi closed intervals.

Rules pertaining to operations on Inequalities:

(i) When any number is added or subtracted from both sides of an inequality, the sign of inequality remains same.
If 5x – 4 > 4x –1, we can surely say that 5x – 4x > –1 + 4 i.e. x > 3 because we are essentially adding 4 and subtracting 3x from both sides. Thus one can transpose terms from one side to other side by changing their signs and the inequality sign will remain the same

(ii) if both sides of an inequality are multiplied or divided by the same number, the inequality sign does not always remain the same. If the number multiplied or divided with is positive, the sign remains the same but if the number is negative the inequality sign reverses.


Thus xy>34${\displaystyle \frac{x}{y}>\frac{3}{4}}$ cannot be restated as   4x > 3y. This is because here we are multiplying both sides by 4y and since we do not know the sign of 4y, we cannot be sure of the inequality remaining the same. For more proof, though −2−1>34${\displaystyle \frac{-2}{-1}>\frac{3}{4}}$  , yet –8 > –3 is wrong.

Thus when we cross multiply, we have to be careful and should know the sign of the expression we are multiplying with. In the above example if we knew that x is negative, we can infer that y is negative (x/y is positive and if x is negative, y has to be negative) and then we can be sure that 4x < 3y. Note that the inequality sign has changed as we are multiplying with 4y which is negative.