Resources for Graphical Solutions of Equations and Inequations
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Graphical Solutions of Equations and Inequations Theory
![Equations and Inequations can be solved algebraically or graphically.\\ \begin{multicols}{2} \textbf{Example 1}\\ Given the functions \(y=x^2-2 x\) and \(y=12-x\), solve the equation \(x^2-2 x=12-x\).\\ \textbf{Example 1 solution}\\ $\begin{aligned} & y=x^2-2 x \ldots(1) \\ & y=12-x \ldots(2) \\ & x^2-2 x=12-x \\ & x^2-x-12=0 \\ & (x-4)(x+3)=0 \\ & \therefore x=4, x=-3 \end{aligned}$\\ From the graph also \(x=4, x=-3\)\\ \begin{center} \begin{tikzpicture}[ declare function={a(\x)=(\x)^2-2*\x;}, declare function={b(\x)=12-\x;}, ] \def \domain{-3.6:5.6} \def \xmax{6} \def \xmin{-6} \def \ymax{20} \def \ymin{-2} \def \xlabel{x} \def \ylabel{y} \begin{axis}[ axis lines=middle, axis line style={Stealth-Stealth,very thick}, grid=both, %major, ylabel = $\ylabel$, xlabel = $\xlabel$, width=3in, height=2.5in, ymin=\ymin, ymax=\ymax, xmin=\xmin, xmax=\xmax, minor x tick num=1, minor y tick num=1, axis line style = thick, major tick style = thick, minor tick style = thick, xtick distance = 1, xlabel style={right}, ytick distance = 2, ylabel style={above}, x grid style={thin, opacity=0.8}, y grid style={thin, opacity=0.8}, axis on top=false, xtick={-6,-4,...,6}, ytick={-5,0,5,10,15,20}, % extra x ticks={0}, extra x tick style={xticklabel style={anchor=north east}} ] %FUNCTION \draw [draw=black,thin, opacity=0.5] (\xmin,\ymin) rectangle (\xmax,\ymax); \addplot[name path=a, ultra thick, latex-latex, samples=300, smooth, domain=\domain, red] {a(x)} node [pos=0.9, left, red, font=\small] {}; \addplot[name path=b, ultra thick, latex-latex, samples=300, smooth, domain=-5.5:5.5, blue] {b(x)} node [pos=0.9, left, blue, font=\small] {}; \node (11) at (-3,15) {}; \node (22) at (4,8) {}; \node (1) [draw ,rectangle, align=center,fill=gray!20] at (-4,10) {\((-3,15)\)}; \node (2) [draw ,rectangle, align=center,fill=gray!20] at (3,13) {\((4,8)\)}; \end{axis} \path[draw,-stealth,font=\ttfamily,line width = 0.5mm] (1)--(11); \path[draw,-stealth,font=\ttfamily,line width = 0.5mm] (2)--(22); \end{tikzpicture} %\includegraphics[width=0.3\textwidth]{Picture1} \end{center} \columnbreak \textbf{Example 2}\\ Given the functions \(y=x\) and \(y=6-x^2\), solve the inequality \(x \leqslant 6-x^2\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} & y=x\cdots \text { (1) } \\ & y=6-x^2\cdots(2) \\ & x=6-x^2 \\ & x^2+x-6=0 \\ & (x-2)(x+3)=0 \\ & \therefore x=2 \text { or } x=-3 \end{aligned}$\\ From the graph\\ \(-3 \leqslant x \leq 2\)\\ \begin{center} \begin{tikzpicture}[ declare function={a(\x)=6-(\x)^2;}, declare function={b(\x)=\x;}, ] \def \domain{-3.4:3.4} \def \xmax{6} \def \xmin{-6} \def \ymax{8} \def \ymin{-6} \def \xlabel{x} \def \ylabel{y} \begin{axis}[ axis lines=middle, axis line style={Stealth-Stealth,very thick}, grid=both, %major, ylabel = $\ylabel$, xlabel = $\xlabel$, width=3in, height=2.5in, ymin=\ymin, ymax=\ymax, xmin=\xmin, xmax=\xmax, minor x tick num=1, minor y tick num=1, axis line style = thick, major tick style = thick, minor tick style = thick, xtick distance = 1, xlabel style={right}, ytick distance = 2, ylabel style={above}, x grid style={thin, opacity=0.8}, y grid style={thin, opacity=0.8}, axis on top=false, xtick={-6,-4,...,6}, ytick={-6,-4,-2,...,8}, % extra x ticks={0}, extra x tick style={xticklabel style={anchor=north east}} ] %FUNCTION \draw [draw=black,thin, opacity=0.5] (\xmin,\ymin) rectangle (\xmax,\ymax); \addplot[name path=a, ultra thick, latex-latex, samples=300, smooth, domain=\domain, red] {a(x)} node [pos=0.9, left, red, font=\small] {}; \addplot[name path=b, ultra thick, latex-latex, samples=300, smooth, domain=-5:5.5, blue] {b(x)} node [pos=0.9, left, blue, font=\small] {}; \draw[latex-latex,line width=1pt](-3,-6)--(-3,8); \draw[latex-latex, line width=1pt](2,-6)--(2,8); \end{axis} \end{tikzpicture} %\includegraphics[width=0.3\textwidth]{Picture2} \end{center} \end{multicols}](/media/4nlfw1km/4774.png)