Resolva, em $\mathbb{R}$, as seguintes inequações

Funções racionais: Aleph 11 - Volume 2 Pág. 50 Ex. 3

by AMMA · Published 5 de Fevereiro de 2014 · Updated 26 de Dezembro de 2021

Enunciado

Resolva, em $\mathbb{R}$, as seguintes inequações:

$\frac{{x + 1}}{{x – 2}} > 0$
$\frac{{ – 5}}{{1 – 2x}} < 0$
$\frac{2}{{{x^2} + 2x}} – \frac{{x + 1}}{{x + 2}} < 0$

Resolução

Tem-se sucessivamente:
\[\begin{array}{*{20}{l}}
{\frac{{x + 1}}{{x – 2}} > 0}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{\left\{ {\begin{array}{*{20}{l}}
{x + 1 < 0} \\
{x – 2 < 0}
\end{array}} \right.}& \vee &{\left\{ {\begin{array}{*{20}{l}}
{x + 1 > 0} \\
{x – 2 > 0}
\end{array}} \right.}
\end{array}} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{\left\{ {\begin{array}{*{20}{l}}
{x < – 1} \\
{x < 2}
\end{array}} \right.}& \vee &{\left\{ {\begin{array}{*{20}{l}}
{x > – 1} \\
{x > 2}
\end{array}} \right.}
\end{array}} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{x < – 1}& \vee &{x > 2}
\end{array}} \\
{}& \Leftrightarrow &{x \in \left] { – \infty , – 1} \right[ \cup \left] {2, + \infty } \right[}
\end{array}\]
Tem-se sucessivamente:
\[\begin{array}{*{20}{l}}
{\frac{{ – 5}}{{1 – 2x}} < 0}& \Leftrightarrow &{1 – 2x > 0} \\
{}& \Leftrightarrow &{x < \frac{1}{2}} \\
{}& \Leftrightarrow &{x \in \left] { – \infty ,\frac{1}{2}} \right[}
\end{array}\]
Tem-se sucessivamente:
\[\begin{array}{*{20}{l}}
{\frac{2}{{{x^2} + 2x}} – \frac{{x + 1}}{{\mathop {x + 2}\limits_{\left( x \right)} }} < 0}& \Leftrightarrow &{\frac{{2 – {x^2} – x}}{{x\left( {x + 2} \right)}} < 0} \\
{}& \Leftrightarrow &{\frac{{{x^2} + x – 2}}{{x\left( {x + 2} \right)}} > 0} \\
{}& \Leftrightarrow &{\frac{{\left( {x – 1} \right)\left( {x + 2} \right)}}{{x\left( {x + 2} \right)}} > 0} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{\frac{{\left( {x – 1} \right)}}{x} > 0}& \wedge &{x \ne – 2}
\end{array}} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{\left( {\begin{array}{*{20}{c}}
{\left\{ {\begin{array}{*{20}{l}}
{x – 1 < 0} \\
{x < 0}
\end{array}} \right.}& \vee &{\left\{ {\begin{array}{*{20}{l}}
{x – 1 > 0} \\
{x > 0}
\end{array}} \right.}
\end{array}} \right)}& \wedge &{x \ne – 2}
\end{array}} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{\left( {\begin{array}{*{20}{c}}
{\left\{ {\begin{array}{*{20}{l}}
{x < 1} \\
{x < 0}
\end{array}} \right.}& \vee &{\left\{ {\begin{array}{*{20}{l}}
{x > 1} \\
{x > 0}
\end{array}} \right.}
\end{array}} \right)}& \wedge &{x \ne – 2}
\end{array}} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{\left( {\begin{array}{*{20}{c}}
{x < 0}& \vee &{x > 1}
\end{array}} \right)}& \wedge &{x \ne – 2}
\end{array}} \\
{}& \Leftrightarrow &{x \in \left] { – \infty , – 2} \right[ \cup \left] { – 2,0} \right[ \cup \left] {1, + \infty } \right[}
\end{array}\]