Rational Inequalities Calculator - Practice Questions & MCQ

We consider the algebraic inequalities of the following types

$\begin{aligned} & \frac{p(x)}{q(x)}<0, \frac{p(x)}{q(x)}>0 \\ & \frac{p(x)}{q(x)} \leq 0, \frac{p(x)}{q(x)} \geq 0\end{aligned}$

If p(x) and q(x) can be resolved in factor then we can solve these types of inequalities using a wavy curved method otherwise we use the following method to solve them.

$
\text { (1) } \begin{aligned}
& \frac{\mathrm{p}(\mathrm{x})}{\mathrm{q}(\mathrm{x})} \\
\Rightarrow & >0 \Rightarrow \mathrm{p}(\mathrm{x}) \mathrm{q}(\mathrm{x})>0 \\
\mathrm{p}(\mathrm{x}) & >0, \mathrm{q}(\mathrm{x})>0 \text { or } \mathrm{p}(\mathrm{x})<0, \mathrm{q}(\mathrm{x})<0
\end{aligned}
$

(2)

$
\begin{aligned}
& \frac{\mathrm{p}(\mathrm{x})}{\mathrm{q}(\mathrm{x})}<0 \Rightarrow \mathrm{p}(\mathrm{x}) \mathrm{q}(\mathrm{x})<0 \\
\Rightarrow & \mathrm{p}(\mathrm{x})>0, \mathrm{q}(\mathrm{x})<0 \text { or } \mathrm{p}(\mathrm{x})<0, \mathrm{q}(\mathrm{x})>0
\end{aligned}
$

(3)

$
\begin{aligned}
& \frac{\mathrm{p}(\mathrm{x})}{\mathrm{q}(\mathrm{x})} \geq 0 \Rightarrow \mathrm{p}(\mathrm{x}) \mathrm{q}(\mathrm{x}) \geq 0 \text { and } \mathrm{q}(\mathrm{x}) \neq 0 \\
& \Rightarrow \mathrm{p}(\mathrm{x}) \geq 0, \mathrm{q}(\mathrm{x})>0 \text { or } \mathrm{p}(\mathrm{x}) \leq 0, \mathrm{q}(\mathrm{x})<0
\end{aligned}
$

$
\text { (4) } \begin{aligned}
& \frac{\mathrm{p}(\mathrm{x})}{\mathrm{q}(\mathrm{x})} \leq 0 \Rightarrow \mathrm{p}(\mathrm{x}) \mathrm{q}(\mathrm{x}) \leq 0 \text { and } \mathrm{q}(\mathrm{x}) \neq 0 \\
\Rightarrow \mathrm{p}(\mathrm{x}) & \geq 0, \mathrm{q}(\mathrm{x})<0 \text { or } \mathrm{p}(\mathrm{x}) \leq 0, \mathrm{q}(\mathrm{x})>0
\end{aligned}
$