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Graphing inequalities is a method we use to visually represent possible solutions to a system of inequalities. When graphing each inequality, you will sketch either solid or dashed lines on a coordinate plane. These lines represent the boundary of the inequality.

• Solid lines are used for inequalities like \( \leq \) or \( \geq \), indicating that the points on the line are part of the solution.
• Dashed lines are used for \( < \) or \( > \) inequalities, showing that points on the line are not included within the solution.

Once lines are drawn, you will "shade" one side of each line. This shading represents the area where the inequality holds true. By graphing several inequalities, you can see the regions where all conditions are satisfied simultaneously.

The solution set of a system of inequalities is the region where the solutions to all individual inequalities overlap on the graph. This set represents all possible solutions that simultaneously satisfy each inequality in the system. For the system at hand:

• The inequality \( y \leq x^2 \) covers the area beneath the parabola, including the boundary.
• The inequality \( y < -x + 6 \) hides the region below the line \( y = -x + 6 \), but not the boundary itself.
• The inequality \( y > x - 6 \) encompasses the area above the line \( y = x - 6 \), excluding the boundary.

After graphing, the solution set is the common shaded area where all three conditions meet. It shows every point (\(x, y\)) that solves the system.

Intersection points are key to understanding a solution set's shape because they define the vertices of the region where inequalities overlap. To find these coordinates, solve for where the boundaries of the inequalities intersect.1. **Parabola and Line Intersection (\( y = x^2 \) and \( y = -x + 6 \))**: Set equations equal to each other, \( x^2 = -x + 6 \). Solving gives \( x = 2 \) and \( x = -3 \), leading to points \((2, 4)\) and \((-3, 9)\).2. **Parabola and Line Intersection (\( y = x^2 \) and \( y = x - 6 \))**: Solve \( x^2 = x - 6 \) using the quadratic formula to find other intersections.These points are crucial as they help delineate the edges of the solution region, acting as its corners or vertices.

A bounded region is one that is encapsulated within a finite area on a graph. In contrast, unbounded regions extend infinitely in one or more directions. To determine if a solution set is bounded:- Look at the nature of each inequality's graph. Parabolas, like \( y \leq x^2 \), can create unbounded regions because they open infinitely.- Lines, depending on their orientation and intersections, can either enclose an area or leave it open-ended.For this system, since the parabola \( y \leq x^2 \) continues indefinitely outwards, the solution set cannot be completely enclosed or bounded. Thus, despite intersections creating a distinct region, the overall solution set remains unbounded.