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In graphing inequalities, the feasible region is the area where all the inequalities in the system overlap. This is crucial because it shows all possible solutions to the set of linear inequalities. To find this region, you first graph each individual inequality. For example, consider the inequalities: \[ x + 3y \leq 15, \ 2x + y \leq 15, \ x \geq 0, \ y \geq 0 \].

Once you have each boundary line plotted, the feasible region is the section where all shaded areas intersect. This region represents where all conditions of the inequalities are simultaneously satisfied. It is essential to accurately identify this region because it encompasses all solutions that meet all the given constraints.

Boundary lines are created by converting inequalities into equalities. They act as the framework for understanding where the inequalities lie. To graph \(x + 3y \leq 15\), you start by plotting its boundary line, which is \(x + 3y = 15\). Find key points like the intercepts to draw the line. The x-intercept is at \(x = 15\) (where y=0), and the y-intercept is at \(y = 5\) (where x=0).

Next, for \(2x + y \leq 15\), the boundary line is \(2x + y = 15\), with x-intercept at \(x = 7.5\) (where y=0) and y-intercept at \(y = 15\) (where x=0).

The inequalities \(x \geq 0\) and \(y \geq 0\) are simple, representing the non-negative parts of their respective axes. These form the horizontal and vertical confines of the feasible region in the first quadrant.

It's important to note the inequality type (\(\leq\) or \(\geq\)) because it determines which side of the boundary line is included in the feasible region.

Vertices of the feasible region are the intersection points of the boundary lines. These points are essential because they define the limits of the feasible region. To find them, solve the equations of the boundary lines simultaneously.

For instance, solving \(x + 3y = 15\) and \(2x + y = 15\) simultaneously, you get the intersection point \((3, 4)\). This involves solving two linear equations:

1. \(x + 3y = 15\)
2. \(2x + y = 15\)

There are other key vertices such as intercepts and origin points like \((0, 0)\), \((7.5, 0)\), and \((0, 5)\). By marking these points on the graph, the feasible region can be precisely determined and shaded, showing the complete set of feasible solutions.

A system of inequalities involves multiple inequality constraints that need to be satisfied simultaneously. Each inequality can be visualized as a shaded region in a coordinate system. The system of inequalities we're considering is:

\[ x + 3y \leq 15, \ 2x + y \leq 15, \ x \geq 0, \ y \geq 0 \]

When graphing a system, start by plotting the boundary line of each inequality. Then determine the feasible region by identifying where all the shaded regions overlap. Each boundary line frames the limits of this overlap.

Systems of inequalities are used in various real-life applications such as financial planning, resource allocation, and optimization problems where multiple constraints are involved, making it vital to understand how to graph and interpret them.