Inequalities Important Formulas, Definitions, & Examples:

Overview:

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.