A system of linear inequalities is a collection of two or more linear inequalities, or equations with two or more variables, that have been grouped together. Graphically, the solution to a system of linear inequalities can be represented as a region in the coordinate plane. In this article, we will explore how to identify which graph shows the solution to the system of linear inequalities.
Graphical Representation
The graph of a system of linear inequalities is a collection of all points that satisfy each of the inequalities in the system. A visual representation of the solution can be obtained by shading the region that is defined by the inequalities. This shading may be done in two different ways: the first is to shade the region that satisfies all of the inequalities, and the second is to shade the region that does not satisfy any of the inequalities.
The shaded region that satisfies all of the inequalities is known as the feasible region and is the solution to the system of linear inequalities. This region is bounded by the lines that represent the inequalities and may contain multiple disconnected regions. The region that does not satisfy any of the inequalities is known as the infeasible region and should not be shaded.
Solving the System
When solving a system of linear inequalities, it is important to remember that any point that satisfies all of the inequalities is a solution. To identify which graph shows the solution to the system of linear inequalities, it is necessary to examine each inequality and determine which region it defines.
For example, if the system contains the inequality “x ≥ 0”, then the region that satisfies this inequality is the region to the right of the y-axis. Once all of the inequalities have been examined, the solution to the system of linear inequalities is the region that is defined by all of the inequalities. This region should be shaded in the graph.
In summary, the solution to a system of linear inequalities can be represented graphically as a region in the coordinate plane. To identify which graph shows the solution to the system of linear inequalities, it is necessary to examine each inequality and determine which region it defines. Once all of the inequalities have been examined, the solution is the region that is defined by all of the inequalities. This region should be shaded in the graph.
