Plotting area in the plane where two inequalities in two variables hold

Question

I would like to plot the area in the plane where two inequalities hold, both involving two variables. I would like to do it the way Wolfram Alpha does it, but within a LaTeX document. I have looked into the documentation of pgfplots, but I cannot figure out if and how this is possible. Maybe I need a different package, but I have had little success with google.

Answer 1

There is a package for LaTeX called sagetex which gives you the power of a CAS in your latex documents. I didn't see it mentioned, and since the example you gave involved something more complicated than linear inequalities you might be interested. It's great for problems involving complicated mathematics like what you see on Wolfram Alpha. Look into Sage; this page was especially helpful to me. Include the sagetex package in the preamble:

\documentclass{article}
\usepackage{sagetex}
\begin{document}
\pagestyle{empty}
\begin{sagesilent}
G=Graphics()
H=Graphics()
I=Graphics()
J=Graphics()
var('x,y')
G+=implicit_plot(2*y^2*x-5*y^2+8*y-4*x*y+3*x-2, (x,-2,4), (y,-2,7),color='red')
H+= implicit_plot(3*y^2*x-8*y^2+12*y-6*x*y+4*x-3,(x,-2,4),(y,-2,7),color='black')
I += region_plot([2*y^2*x-5*y^2+8*y-4*x*y+3*x-2>=0, 3*y^2*x-8*y^2+12*y-6*x*y+4*x-3<=0], (x,0,6), (y,-2,7),plot_points=400,frame=True,axes=False)
J += text("$\chi$", (4, 1), fontsize=16, color='black')
\end{sagesilent}
\[ \sageplot{I+H+G}\] 
\[ \sageplot{I+J}\] 
\end{document}

This gave me output:

The first plot has the implicit plots drawn in, the second just has the region. I added a latex character, \chi, to the diagram to show you that you can add LaTeX text onto your figure. Changing \sageplot{I+J} to just \sageplot{I} would remove it. By modifying the code you ought to be able to get the output you want and have it look even better than Wolfram Alpha.