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In inequalities, the solution set refers to the collection of all possible solutions that satisfy the conditions posed by the inequalities. This solution set can sometimes be bounded or unbounded. When it's bounded, it means the solution set forms a closed and limited area on the graph, with no part extending into infinity.
In contrast, unbounded solution sets extend indefinitely in one or more directions. For the system of inequalities in our exercise, the solution set forms a trapezoidal shape on the graph, which does not stretch infinitely.
Hence, it is considered a bounded solution set. This closure occurs because all inequalities intersect and form a closed shape without any line going on forever.

Linear inequalities are expressions that show a relationship of inequality between two algebraic expressions. They are quite similar to linear equations, except instead of using an equal sign, they use inequality signs like \(\leq\), \(<\), \(\geq\), or \(>\).
Essentially, they define a half-plane on a coordinate plane. For example, the inequality \(x + y \leq 6\) includes all the points that lie on or below the line \(x + y = 6\).
Linear inequalities are essential in defining feasible regions on a graph, helping depict areas of interest such as constraints in optimization problems or boundary definitions in resource allocation.

Graphing inequalities involves plotting the inequality on a coordinate plane and identifying the region of the graph that satisfies the inequality condition.
To do this, you start by graphing the equation as if it had an equal sign, which gives the boundary line of the inequality. Once the boundary line is drawn, you determine which side of the line satisfies the inequality by testing a point.
For instance, if one inequality is \(x + y \leq 6\), plot the line \(x + y = 6\). Then, shade the area below this line because it represents all points \((x,y)\) where their sum is less than or equal to 6.
Continuing with other inequalities in the system, you proceed through the same process, determining the feasible region by shading appropriately.

The intersection of feasible regions is the area where all the shaded areas from different inequalities overlap. This intersection represents all solutions that satisfy every inequality in the system simultaneously.
It is crucial because it gives you exactly where you can find valid solutions that meet all constraints defined by the inequalities.
In the provided exercise, after graphing and shading the regions for each inequality, the overlapping region or intersection forms a trapezoid. This indicates that all points within this trapezoidal shape satisfy the system of inequalities.
Carefully identifying this intersection is important for solving systems of inequalities, especially in fields like linear programming and operations research where optimal solutions are sought within constrained regions.