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When we talk about linear inequalities, we're discussing expressions where two linear expressions are compared using inequality signs like >, <, >=, and <=. Unlike equalities (equations), which denote exact values, inequalities allow for a range of values. For instance, the inequality 3x + 2y >= 6 tells us that the sum of three times x and two times y is greater than or equal to 6. In the system of inequalities from our exercise, we're dealing with multiple linear inequalities that together define a common solution region.

To visualize these solutions, we need to express each inequality in terms of y, allowing us to graph them in a two-dimensional coordinate system. The goal is to identify the region of the plane where all inequalities hold true simultaneously.

The process of graphing inequalities is a straightforward extension of graphing linear equations. With an inequality, instead of drawing a line to represent an equation, we shade an entire region of the graph. This shaded area represents all possible solutions to the inequality. For instance, when graphing y >= 30 - 3x/2, you would first graph the line y = 30 - 3x/2 as if it were an equation. Then, since the inequality is >=, you would shade the area above the line. This indicates that every point in that shaded area is a solution. When working with a system of inequalities, we graph each one on the same coordinate plane and look for the overlapping shaded areas, which indicate where all the inequalities are satisfied simultaneously.

Remember, the inequality sign tells us which side of the line to shade. For > or >=, we shade above the line. For < or <=, we shade below. Also, if the inequality is strict (> or <), the line is dashed to show that points on the line itself are not included in the solution.

When graphing the system of inequalities, we determine whether the solution set is a bounded region or an unbounded one. A bounded region has a finite area, entirely enclosed by lines or curves, and it has edges that we can trace. In the context of our exercise, where all inequalities are in the first quadrant and intersect with each other and the axes, the overlapping shaded areas create a polygon. This polygon is the bounded region with a definite boundary. An unbounded region, in contrast, extends infinitely in at least one direction and thus does not form a closed shape. The bounded nature of the region is particularly important when solving optimization problems because it guarantees that a maximum or minimum value exists within the region.

The corner points (also known as vertices) of a bounded region are points of intersection where the boundary lines meet. They are crucial in optimization problems because, according to the linear programming principle, the optimal solution lies at one of these corner points. To find them, we solve a system of equations made from each pair of boundary lines. In our exercise, we calculate where each pair of lines intersects by setting the equations equal to each other and solving for x and y. Once we find the corner points, we can evaluate any objective function at these points to find the optimal solution.

From our step by step solution, we found that the corner points were at (12, 18), (18, 12), and (24, 10). Determining these points allows us to proceed with related tasks, such as optimization or further analysis of the feasible region.