\documentclass{article}

\usepackage{mathtools}
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\title{Analyse 1 - TD 1 \\ NON CORRIGÉ}
\author{Timéo Pochin}

\begin{document}
    \maketitle

    \section*{Exercice 1}

    \subsection*{a)}
    \begin{align}
              & 5x+2\geq-3 \nonumber \\
    \iff\quad & 5x\geq-5 \nonumber \\
    \iff\quad & x\geq-1 \nonumber \\
    \iff\quad & x=\left[-1,+\infty\right[ \nonumber
    \end{align}

    \subsection*{b)}
    \begin{align}
              & 2x-1<4x+3\leq-x+6 \nonumber \\
    \iff\quad & (2x-1<4x+3)\land(4x+3\leq-x+6) \nonumber \\
    \iff\quad & (-4<2x)\land(5x\leq 3) \nonumber \\
    \iff\quad & (-2<x)\land(x\leq\frac{3}{5}) \nonumber \\
    \iff\quad & x=\left]-2,\frac{3}{5}\right] \nonumber
    \end{align}

    \subsection*{c)}
    \begin{align}
              & |x-1|<4 \nonumber \\
    \iff\quad & (x-1<4)\land(x-1>-4) \nonumber \\
    \iff\quad & (x<5)\land(x>-3) \nonumber \\
    \iff\quad & x=\left]-3,5\right[ \nonumber
    \end{align}

    \subsection*{d)}
    \begin{align}
              & |x-2|\geq 3 \nonumber \\
    \iff\quad & (x-2\geq 3)\lor(x-2\leq-3) \nonumber \\
    \iff\quad & (x\geq 5)\lor(x\leq-1) \nonumber \\
    \iff\quad & x=\left]-\infty,-1\right]\ \cup\ \left[5,+\infty\right[ \nonumber
    \end{align}

    \subsection*{e)}
    \begin{align}
              & |x-2|\leq|x| \nonumber \\
    \iff\quad & (x-2)^2\leq x^2 \nonumber \\
    \iff\quad & x^2-4x+4\leq x^2 \nonumber \\
    \iff\quad & 4\leq 4x \nonumber \\
    \iff\quad & 1\leq x \nonumber \\
    \iff\quad & x=\left[1,+\infty\right[ \nonumber
    \end{align}

    \subsection*{f)}

    \subsection*{g)}
    \begin{align}
              & \sqrt{x+1}<2 \nonumber \\
    \iff\quad & (x+1<4)\land(x+1\geq0) \nonumber \\
    \iff\quad & (x<3)\land(x\geq-1) \nonumber \\
    \iff\quad & x=\left[-1,3\right[ \nonumber
    \end{align}

    \subsection*{h)}
    \begin{align}
              & x^2+1\leq 3 \nonumber \\
    \iff\quad & x^2\leq 2 \nonumber \\
    \iff\quad & (x\leq\sqrt{2})\land(x\geq-\sqrt{2}) \nonumber \\
    \iff\quad & x=\left[-\sqrt{2},\sqrt{2}\right] \nonumber
    \end{align}

    \subsection*{i)}
    \begin{align}
              & x^2+3x<4 \nonumber \\
    \iff\quad & x^2+3x-4<0 \nonumber \\
    \iff\quad & (x-1)(x+4)<0 \nonumber \\
    \iff\quad & (x<1)\land(x>-4) \nonumber \\
    \iff\quad & x=\left]-4,1\right[ \nonumber
    \end{align}

    \subsection*{j)}
    \begin{align}
              & x^3-3x^2+2x\geq 0 \nonumber \\
    \iff\quad & x(x^2-3x+2)\geq 0 \nonumber \\
    \iff\quad & x(x-1)(x-2)\geq 0 \nonumber \\
    \iff\quad & \big((x\geq 0)\land(x\leq 1)\big)\lor(x\geq 2) \nonumber \\
    \iff\quad & x=\left[0,1\right]\cup\left[2,+\infty\right[ \nonumber
    \end{align}

    \subsection*{k)}

\end{document}
