Graph each system of linear inequalities. State whether the...

Graph each system of linear inequalities. State whether the graph is bounded or unbounded, and label the corner points. $$\left\{\begin{array}{r}x \geq 0 \\y \geq 0 \\2 x+y \leq 6 \\x+2 y \leq 6\end{array}\right.$$

Key Concepts

Bounded vs Unbounded Regions

A bounded region is completely enclosed by its boundaries and is limited in size, while an unbounded region extends infinitely in at least one direction. Determining whether a feasible region is bounded or unbounded is crucial in understanding the nature of the solution set.

Linear Inequalities

Linear inequalities are mathematical expressions that use inequality symbols instead of equality. They define a range of possible values for variables, and when combined as a system, they restrict the solution set to the points that satisfy every inequality in the set.

Feasible Region

The feasible region is the set of all points that meet every condition stated by the linear inequalities. This region is typically shown on a graph where overlapping shaded areas represent the solutions that satisfy the system as a whole.

Graphing Techniques

To graph linear inequalities, one typically starts by drawing the boundary line represented by replacing the inequality with an equality. Then, using a test point such as the origin, one identifies which side of the line satisfies the inequality and shades the appropriate segment of the plane.

Corner Points (Vertices)

Corner points, or vertices, are the points where the boundary lines of the inequalities intersect. These points are important in various mathematical applications, including optimization, because extreme values of certain functions often occur at these intersections.

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