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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Author: Gregory V. Bard. Sage for Undergraduates. American Mathematical Society, Providence, RI, 2015. Available at http://www.gregorybard.com/Sage.html.

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Date: 02 Mar 2019 22:01

Submitted by: Zane Corbiere

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