Rational Inequalities

Inequalities are something where Functions are compared involving sign of inequality. These points can be represented in the number line. The inequality is represented as <,>,$\leq$ and $\geq$.
Consider a rarional function f(x) = $\frac{p}{q}$, where p and q are integers and q $\neq$ 0 is said to be function of rational inequality iff

A number line can be thought of a line extending from negative infinity to positive infinity, with zero as center.

Solve the following rational inequality:
$\frac{x^{2} + 7x + 12}{x + 6}$ $\geq$ 0
$\frac{x^{2} + 7x + 12}{x + 6}$  =  $\frac{(x + 4)(x + 3)}{(x + 6)}$
The polynomial fraction will be zero where numerator is zero so equate it to zero.
(x² + 7x + 12) = 0
or
(x + 4)(x + 3) = 0
The Zero of (x + 4) is -4
The Zero of (x + 3) is -3.
Set the value of denominator to zero we get x = -6.
we get three zeros namely -6,-4 and -3 which can divide the number line into four intervals (-$\infty$, -6) (-6, -4) (-4, -3) and (-3, $\infty$).



$\frac{x + 1}{x -1}$ > 0
The polynomial fraction will be zero where numerator is zero so equate it to zero.
x + 1 = 0
x = -1
The Zero of (x + 1) is -1
The polynomial of denominator should also be Set to zero
x - 1 = 0
x = 1
The Zero of (x - 1) is -4
we get two zeros namely -1 and -4 which can divide the number line into three intervals (- $\infty$, -1) (-1,-4) and (-4, $\infty$).