Originally Posted by

**ticbol** This is good for practice.

It will be long if I show it all, so let me assume you know how to graph inequalities and how to find the intersection of two inequalities.

I don't know how to sketch figures here so sketch it on paper.

Originally Posted by

**Soroban**

As ticbol pointed out, the region is a quadrilateral.

But I differ on one vertex.

$\displaystyle [1] \cap [2]\!:\;\left(\frac{10}{3},\,\frac{2}{3}\right)$

$\displaystyle [1] \cap [4]\!:\;\left(1,\,\frac{8}{5}\right)$

$\displaystyle [2] \cap [3]\!:\;(4,\,2)$

$\displaystyle [3] \cap [4]\!:\;(1,\,5)$

If you want to draw graphs badly enough, the picture environment isn't too awful. Here is the quadrilateral.

$\displaystyle \setlength{\unitlength}{1cm}

\begin{picture}(4,4)

\qbezier(3.33,.667)(3.33,.667)(1,1.6)

\qbezier(4,2)(4,2)(3.33,.667)

\qbezier(4,2)(4,2)(1,5)

\qbezier(1,1.6)(1,1.6)(1,5)

\qbezier(0,0)(0,0)(4,0)

\qbezier(0,0)(0,0)(0,5)

\end{picture}

$

Below is the LaTeX code. I have something like this stored in a document. I cut and pasted it here and then modified it.

The lines are drawn by qbezier. \qbezier(X,Y)(X,Y)(V,W) draws a line beteen points (X,Y) and (V,W). Note the (X,Y) is repeated. You could also repeat the (V,W) with same effect and the order of the points does not matter.

The \setlength, \begin{picture} and \end{picture} commands are just a little housekeeping that needs to be done. The (4,4) in \begin{picture} sets the size of the axes. The \setlength sets the size of the picture. If LaTeX complains about the size of the image, reduce the length.

Code:

\setlength{\unitlength}{1cm}
\begin{picture}(4,4)
\qbezier(3.33,.667)(3.33,.667)(1,1.6)
\qbezier(4,2)(4,2)(3.33,.667)
\qbezier(4,2)(4,2)(1,5)
\qbezier(1,1.6)(1,1.6)(1,5)
\qbezier(0,0)(0,0)(4,0)
\qbezier(0,0)(0,0)(0,5)
\end{picture}

The first 4 qbeziers draw the quadrilateral. I just put the coordinates in straight from Soroban's post. The second 2 qbeziers draw the axes.