Graph the system of inequalities. Tell whether the graph is...

Graph the system of inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$$
\left\{\begin{array}{c}{x \geq 0} \\ {y \geq 0} \\ {x+y \leq 6} \\ {2 x+y \leq 10}\end{array}\right.
$$

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Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting lines corresponding to the equality part of an inequality and then determining which side of the line satisfies the inequality. This method is used to visually represent constraints on variables in the coordinate plane.

Feasible Region

The feasible region is the set of all points that satisfy every inequality in a system. It is typically found by graphing all the inequalities and identifying the common overlapping area. This region represents all possible solutions to the constraint system.

Bounded vs Unbounded Regions

A bounded region is one that is completely enclosed by the boundaries (no direction extends to infinity), while an unbounded region extends infinitely in at least one direction. Determining whether a feasible region is bounded is important for understanding the limits of potential solutions, such as in optimization problems.

Corner Points (Vertices)

Corner points, or vertices, are the intersections of the boundary lines of the feasible region. In linear programming, these points are often critical because they represent potential candidates for an optimal solution. Identifying these points involves solving systems of equations where the boundaries intersect.

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