Solving Systems of Linear Inequalities - Maple Help

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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 $\mathbf{\le }$   is less than or equal to (in Maple this is denoted as <=) $\mathbf{\ge }$   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,  is a linear inequality because x and y have exponents 1. However, 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 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:

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:

 1 Graph the first equation $y\ge x$ by changing the$\ge$for a =. So the equation to be plotted will be $y=x$.
 2 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.
 3 Decide whether to include or not the line $y=x$. As previously mentioned, since the inequality $y\ge x$ is non-strict then you should draw $y=x$ as a solid line.
 4 Repeat this for all the inequalities in the system.
 5 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:

 $\left\{\left[\left\{{x}{\le }{-}\frac{{1}}{{3}}{,}{-5}{<}{x}\right\}{,}\left\{{x}{\le }{y}{,}{y}{<}{2}{}{x}{+}{5}\right\}\right]{,}\left[\left\{{-}\frac{{1}}{{3}}{<}{x}{,}{x}{<}{2}\right\}{,}\left\{{x}{\le }{y}{,}{y}{<}{-}{x}{+}{4}\right\}\right]\right\}$ (3.2.1)

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.

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 to

 yx ><<=>=

 yx ><<=>=

 yx ><<=>=

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