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When we talk about quadratic inequalities, we refer to expressions in the form of a quadratic equation but instead of an equal sign, we have inequalities like <, >, \(\leq\), or \(\geq\). A quadratic inequality such as \(y \geq x^{2}-4\) involves a quadratic expression on one side of the inequality.
To visualize the solutions for this type of inequality, we graph the related quadratic equation \(y = x^{2}-4\). The resulting curve is a parabola. In the inequality \(y \geq x^{2}-4\), the symbol \(\geq\) indicates that we include not only the points on the parabola but also the region above it. This region represents all the \(y\)-values that are greater than or equal to \(x^{2}-4\) for each \(x\)-value.
It's important for students to recognize that the inequality sign determines the direction of the parabola's shading. For \(y \leq x^{2}-4\) we shade downwards, for \(y \geq x^{2}-4\) we shade upwards.

Linear inequalities look similar to linear equations, but again, they have an inequality sign instead of an equal sign. Rather than just a line, linear inequalities represent a half-plane in a two-dimensional space. The inequality \(x-y \geq 2\), or rewritten as \(y \leq x - 2\), describes not just a line but all the points below it.
The line \(y = x - 2\) is graphed with a solid line to show that points on the line are solutions to the inequality. The half-plane below the line represents all ordered pairs \( (x,y)\) that satisfy the inequality \(y \leq x - 2\). Recognizing the importance of the inequality sign and the direction of the inequality determines which side of the line to shade.

The solution set of a system of inequalities consists of all the points that satisfy all inequalities in the system simultaneously. As we work with systems, like the one composed of \(y \geq x^{2}-4\) and \(y \leq x - 2\), our goal is to find where the shaded regions of each individual inequality overlap.
To find this overlapping region, which represents the common solution set, we graph each inequality separately and then look for the intersection of the shaded areas. It’s essential to use a solid line for the boundaries of these regions when the inequalities include the equal sign (\(\geq, \leq\)). All the points in the overlapped area satisfy both inequalities; hence, they form the solution set for the system.

Parabola graphing is a crucial skill when working with quadratic inequalities. A parabola is the graph of a quadratic equation of the form \(y = ax^{2} + bx + c\) and it can open upwards or downwards. The direction of the opening is determined by the coefficient \(a\): if \(a \gt 0\), it opens upwards; if \(a \lt 0\), it opens downwards.
The vertex of the parabola is a significant feature as it represents either the highest or lowest point, depending on the opening direction. In our example, the parabola for \(y = x^{2}-4\) has its vertex at (0,-4) and opens upwards. When graphing a parabola for an inequality, make sure to use a solid curve if the inequality includes the equal sign and consider the direction of shading based on the inequality sign to properly represent the solution set.