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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1315743378150--
%%--Ordre et valeurs absolues--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
$n$ \'{e}tant un entier naturel, comparer les nombres $a$ et $b$ dans les deux
cas suivants:%
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\begin{enumerate}
\item
\[
a=\frac{1}{3}\quad\text{et}\quad b=\frac{50+2n}{74+6n}\text{;}%
\]


\item
\[
a=\frac{n+2}{n+3}\quad\text{et}\quad b=\frac{n+4}{n+5}\text{.}%
\]

\end{enumerate}%

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\end{multicols}%
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\end{exercise}

\begin{exercise}
Soit $a$ un r\'{e}el de l'intervalle $\left]  0;1\right[  .$ En pr\'{e}cisant
votre d\'{e}marche, ranger dans l'ordre croissant les nombres:%
\[
a,\quad a^{2},\quad a^{3},\quad\sqrt{a},\quad\frac{1}{a},\quad1.
\]


\begin{enumerate}
\item Soient $a$ et $b$ deux r\'{e}els tels que: $1\leq a\leq b$;
d\'{e}terminer le signe de $(a-b)(a+b-2)$.

\item Soit $a$ un r\'{e}el tel que: $1\leq a\leq2$; encadrer $p=2a+1$ et
$q=1/2-a/3$.

\item Deux r\'{e}els $x$ et $y$ sont tels que $x/y>1$. Ordonner $x$ et $y:$%
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\begin{enumerate}
\item si $x$ et $y$ sont tous deux positifs;

\item si $x$ et $y$ sont tous deux n\'{e}gatifs.
\end{enumerate}%

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\end{enumerate}
\end{exercise}

\begin{exercise}
Soit $a$, $b$ et $c$ trois réels positifs, montrer que:
$$\sqrt{2}(a+b+c)\leqslant \sqrt{a^2+b^2}+\sqrt{a^2+c^2}+\sqrt{b^2+c^2}$$
\end{exercise}

\begin{exercise}
D\'{e}terminer $I\cap J$ et $I\cup J$ dans chacun des cas suivants:%
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\begin{enumerate}
\item $I=\left[  -3;7\right]  \quad$et\quad$J=\left]  4;+\infty\right[  $;

\item $I=\left]  -1;+\infty\right[  \quad$et\quad$J=\left[  -2/3;+\infty
\right[  $;

\item $I=\left]  \pi/4;3\pi/2\right[  \quad$et\quad$J=\left[  0;\pi/2\right]
$.
\end{enumerate}%

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\end{exercise}

\begin{exercise}
Donner les valeurs de $x$ pour lesquelles on a:%
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\begin{enumerate}
\item $\left\vert x\right\vert =x$;

\item $\left\vert x\right\vert =-x$;

\item $\left\vert x-2\right\vert =2-x$;

\item $\sqrt{(2x-3)^{2}}=2x-3$.
\end{enumerate}%

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\end{exercise}

\begin{exercise}
Transformer chaque encadrement en une in\'{e}galit\'{e} du type $\left\vert
x-c\right\vert \leq R$:%
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\begin{enumerate}
\item $1.45\leq x\leq1.55$;

\item $-4.8\leq x\leq-2.6$;

\item $\pi-0.01\leq x\leq\pi+0.001$.
\end{enumerate}%

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\end{exercise}

\begin{exercise}
On consid\`{e}re la fonction d\'{e}finie par $f(x)=2\left\vert x-3\right\vert
+\left\vert 2x+1\right\vert $.

Exprimer $f(x)$ sans valeurs absolues suivant que:%
\[
x\leq-\frac{1}{2},\quad\text{ou que\quad}-\frac{1}{2}<x<3,\quad\text{ou
que\quad}x\geq3\text{.}%
\]


En d\'{e}duire les solutions de l'\'{e}quation $f(x)=12$.
\end{exercise}


\end{document}