Solving Systems of Linear Inequalities
Main Concept
Linear Inequalities
An inequality is a relation that holds between two values when they are different. The following symbols are used to denote inequality:
< is less than
> is greater than
≤ is less than or equal to (in Maple this is denoted as <=)
≥ is greater than or equal to (in Maple this is denoted as >=)
≠ is not equal to (in Maple this is denoted as <>)
For example, for two values, a and b, when a is less than b, this is denoted as a < b.
A linear inequality is an inequality (containing at least one of the symbols above) in which all of its variables have exponent 1.
For example, x< y is a linear inequality because x and y have exponents 1.
However, x2<y is not a linear inequality because x has exponent 2.
Graphing a Linear Inequality
When a linear inequality is visualized, all of the points where the inequality is satisfied (points lying in the feasible region) are highlighted.
For example, given the inequality x<y one would color in all of the areas on the graph where the x value is less than the y value.
A simple way to graph a linear inequality is to start by substituting in the symbol of inequality with a symbol of equality into an expression and then graph the equation, which will be a line. After that, color in all of the area on the side of the line where the inequality is true.
If x is less than but not equal to y, then the inequality is strict. If x is less than OR equal to y, then the inequality is called non-strict. When a non-strict inequality is plotted, the line of x = y is drawn with a solid line, because x = y is part of the feasible region that solves the non-strict inequality. If the inequality is strict, then the line x = y is drawn with a dotted line because it is not included in the feasible region for the strict inequality.
Systems of Linear Inequalities and How to Solve Them
Mathematical Solution
A system of linear inequalities is a set of linear inequalities as shown in the example below:
y≥x, y<−x+4,y<2⋅x+5
The solution to a system of linear inequalities is the set of all points in the plane that satisfy all of the inequalities in the system.
This solution can be plotted graphically as follows:
Graph the first equation y≥x by changing the≥for a =. So the equation to be plotted will be y=x.
Decide whether to shade the area above the line or below the line. If y is less than x then you should shade the region below the line. If y is greater than x then you should shade the region above since y is greater than or equal to x.
Decide whether to include or not the line y=x. As previously mentioned, since the inequality y≥x is non-strict then you should draw y=x as a solid line.
Repeat this for all the inequalities in the system.
The region of the graph where all the shaded regions intersect is the solution to the system of linear inequalities.
Solution using Maple
To solve a system of linear inequalities with Maple, use the LinearMultivariateSystem command in the SolveTools[Inequality] package.
For example, to solve the system above:
SolveTools:-Inequality:-LinearMultivariateSystemy≥x, y<−x+4,y<2⋅x+5,x,y
x≤−13,−5<x,x≤y,y<2x+5,−13<x,x<2,x≤y,y<−x+4
The description of the command, how to use it and examples is given here.
The graphical solution option for a system of inequalities can also be obtained by using the inequal command in the plots package.
More information on how to use this command is given here.
Instructions: Enter expressions in terms of x into the textboxes, which represent the right sides of equations of linear inequalities. Then press Redraw Graph to view a graphical solution to the input system of linear inequalities.
Plot Options
The default view for the plot is x = -15..15, y=-10..10. Change the view for x and y below:
Range for x
Range for y
to
yx
><<=>=
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