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Graphing inequalities involves visualizing regions on a coordinate plane where the coordinates satisfy the given inequality conditions. It starts with graphing the boundary lines or curves of the inequalities. Before shading the region, determine if the boundary is included (solid line) or not (dashed line). For example, for the inequality \(y > x^2 - 3\), start with graphing the parabola as if it were \(y = x^2 - 3\). Since the inequality is not inclusive (>) and does not include the boundary, use a dashed line.
Similarly, for \(y < x + 1\), graph the boundary line \(y = x + 1\) with a dashed line to show the inequality is strict and does not include the line itself. Then, shade the regions that satisfy each inequality. The region where the shaded areas overlap is the solution set.

Quadratic inequalities involve a variable being squared, forming parabolic boundaries. For instance, \(y > x^2 - 3\) represents a region above the parabola \(y = x^2 - 3\). To graph it, first identify the vertex and the direction it opens. Here, the vertex is at (0, -3) and the parabola opens upward.
Draw a dashed parabola since the boundary is not included (>) and shade above it to represent where \(y\) values are greater than the quadratic expression. Quadratic inequalities can often result in two distinct regions, so carefully check for where the inequality condition holds.
The graph assists in visualizing complex relationships between the variables.

Linear inequalities, like \(y < x + 1\), are simpler to graph compared to quadratic inequalities. They form straight lines upon graphing the boundary equation \(y = x + 1\). Each linear inequality divides the plane into two halves. Graph the line with a dashed method for a non-inclusive (< or >) boundary.
To determine which side to shade, select a test point (like (0,0)), and see if it satisfies the inequality. For \(y < x + 1\), (0,0) tests as follows: \(0 < 0 + 1\) which is true, so shade the region containing (0,0) below the line.
This represents all points where \(y\) values are less than those given by the linear equation.

The solution set of a system of inequalities is the region where the shaded areas of individual inequalities overlap. It can be visualized by graphing each inequality and identifying the intersecting region.
For the system: 1) \(y > x^2 - 3\) and 2) \(y < x + 1\), start by graphing and shading each inequality individually. The solution set is the shared area. This method ensures all variable values in the region meet both conditions.
When dealing with such systems, pay attention to:

• Type of boundary lines (solid for inclusive, dashed for non-inclusive)
• Correct shadings for each inequality
• Intersection of shaded regions.

This approach helps in accurately identifying and graphing the solution set.