Plotting Linear Inequality Systems

## Description

Here we will show how to plot the following system of linear inequalities:

(1)
\begin{align} y &\leq 1.5 - 2x\\ y &\leq 1 - x\\ y &\leq 0.5 - 0.5x\\ x &\geq 0\\ y &\geq 0 \end{align}

Sage has no inequality plotting function, so we will instead plot

(2)
\begin{align} y &= 1.5 - 2x\\ y &= 1 - x\\ y &= 0.5 - 0.5x\\ x &= 0\\ y &= 0 \end{align}

and use the fill, fillcolor and fillalpha commands to represent the inequalities manually. When graphing $x = 0$, we will instead use $y = 10^9x$ to represent a line with a vertical slope, as the normal plot() function cannot plot $x = 0$. Other parameters set are ymin and ymax, to prevent fill from poorly adjusting the viewing window, color to provide more easily visible borders to the regions, and gridlines to increase graph readability.

## Sage Cell

#### Code

p1 = plot(1.5 - 2*x, (x, -0.1, 1.75), ymin=-0.25, ymax=1.75, fill=10, fillcolor='cyan', color='black', fillalpha=1/3, gridlines='minor')
p2 = plot(1 - x, (x, -0.1, 1.75), ymin=-0.25, ymax=1.75, fill=10, fillcolor='magenta', color='black', fillalpha=1/3)
p3 = plot(0.5 - 0.5*x, (x, -0.1, 1.75), ymin=-0.25, ymax=1.75, fill=-10, fillcolor='yellow', color='black', fillalpha=1/3)
p4 = plot(0, (x, -0.1, 1.75), ymin=-0.25, ymax=1.75, fill=-10, fillcolor='gray', color='black', fillalpha=1/3)
p5 = plot((10^9)*x, (x, -0.1, 1.75), ymin=-0.25, ymax=1.75, fill=10, fillcolor='gray', color='black', fillalpha=1/3)
(p1 + p2 + p3 + p4 + p5).show()


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