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When we talk about a system of inequalities, we are referring to multiple inequalities that are considered together. Much like a system of equations, these groups of inequalities have a set of solutions that satisfy all inequalities in the system simultaneously.

The key point is that each inequality can be visualized as a region in a coordinate plane, and when dealing with a system, we are looking for the overlap of these regions. Here, each individual inequality divides the plane into two halves: one that satisfies the inequality and one that does not. The solutions of the system are then found in the areas where these halves intersect.

For instance, a simple system might consist of inequalities like those in the provided exercise – each one contributes to the narrowing down of the feasible solution space. Solving these systems can be especially helpful in real-world applications such as determining constraints for budgeting issues or in designing objects within certain dimensional limits.

Inequality graphing is a visual representation that illustrates the solutions to inequalities on a coordinate plane. Each graphed inequality typically results in a shaded region, which demonstrates all the possible solutions.

To properly graph an inequality, one must first manipulate it into a recognizable form, often resembling the equation of a line. Then, the boundary of the inequality (the line itself) is drawn. If the inequality is strict (< or >), the line is dashed indicating that points on the line are not included as solutions. If it's inclusive (≤ or ≥), the line is solid, including all points on it.

The final and crucial step involves shading: the region of the graph that satisfies the inequality is shaded, which visually captures all the possible solutions. This process is repeated for each inequality in the system, and where the shaded regions overlap is the solution set to the system. In the context of the textbook exercise, graphing each inequality carefully and finding their intersection helps identify the feasible region that satisfies the entire system.

Linear inequalities are similar to linear equations but with inequality signs instead of an equal sign. They describe a range of possible values for variables that make the inequality true. These inequalities can be one-dimensional or two-dimensional, and they can involve one variable (e.g., x < 4) or two (e.g., 2x + 3y > 6).

Graphically, linear inequalities in two variables form half-planes in the coordinate system. The edge of the half-plane, where the inequality changes from true to false, is a straight line, either dashed or solid. The key aspects when graphing include identifying which side of the boundary line to shade and whether to use a dashed or solid line, as mentioned previously.

Once each inequality is graphed, solving the system is all about finding where all these half-planes intersect, which represents all the solutions that satisfy every inequality in the system. The exercise provided helps practice these concepts by requiring a graph that combines several linear inequalities into a comprehensive visual representation of their collective solution set.