# Inequalities Important Formulas, Definitions, & Examples:

#### Overview:

 Topic Included: Formulas, Definitions & Exmaples. Main Topic: Quantitative Aptitude. Quantitative Aptitude Sub-topic: Equations and Inequalities Aptitude Notes & Questions. Questions for practice: 10 Questions & Answers with Solutions.

Inequality: It is the relation between the two different values as given below.

$$a \geq b$$ means that 'a' is equal to and greater than 'b'.

$$a = b$$ means that 'a' is equal to 'b'.

$$a \lt b$$ means that 'a' is less than 'b'.

$$a \gt b$$ means that 'a' is greater than 'b'.

$$a \neq b$$ means that 'a' is not equal to 'b'.

$$a \leq b$$ means that 'a' is equal to and less than 'b'.

Example(1): $$\frac{2}{3} \times \frac{5}{4} \ x = \frac{8}{12} \times \frac{30}{16} \ y$$

Solution: $$\frac{2}{3} \times \frac{5}{4} \ x = \frac{8}{12} \times \frac{30}{16} \ y$$ $$\frac{5}{6} \ x = \frac{5}{4} \ y$$ $$\frac{5}{3} \ x = \frac{5}{2} \ y$$ $$10x = 15y$$ $$\frac{x}{y} = \frac{15}{10}$$ $$x = 15, \ and \ y = 10$$ Hence,$$x \gt y$$

Example(2): (I). $$x^2 = 64$$, (II). $$y = \sqrt{64}$$

Solution: $$x^2 = 64$$ $$x = \pm 8$$ $$y = \sqrt{64}$$ $$y = 8$$ Hence, $$x \leq y$$

Example(3): (I). $$x = \sqrt{81}$$, (II). $$y = (-3)^2$$

Solution: $$x = 9$$ $$y = 9$$ Hence, $$x = y$$

Example(4): (I). $$x + y = 6$$, (II). $$x - y = 8$$

Solution: $$x + y = 6......(1)$$ $$x - y = 8......(2)$$ By adding both equations we get, $$(x + y) + (x - y) = 6 + 8$$ $$x + y + x - y = 14$$ $$2x = 14$$ $$x = 7$$ By putting the value of x in equation (1), we get$$x + y = 6$$ $$7 + y = 6$$ $$y = -1$$ Hence, $$x \gt y$$

Example(5): (I). $$x^2 + 5x - 14 = 0$$, (II). $$y^2 + 3y - 4 = 0$$

Solution: By solving equation (1) we get,$$x^2 + 5x - 14 = 0.....(1)$$ $$x^2 + 7x - 2x - 14 = 0$$ $$x \ (x + 7) - 2 \ (x + 7) = 0$$ $$(x + 7) \ (x - 2) = 0$$ $$x = -7, \ and \ x = 2$$ By solving equation (2) we get,$$y^2 + 3y - 4 = 0.....(2)$$ $$y^2 + 4y - y - 4 = 0$$ $$y \ (y + 4) - 1 \ (y + 4) = 0$$ $$(y + 4) \ (y - 1) = 0$$ $$y = -4, \ and \ y = 1$$ Here, the relation between x and y cannot be established.