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Quadratic inequalities are expressions where a quadratic polynomial is set against a different value in an inequality. A quadratic polynomial generally takes the form of \(ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). Solving quadratic inequalities like the one in our exercise \(2x^{2} - y - 3 > 0\), requires us to understand that we are not finding a single solution, but a range of values that satisfy the inequality.

To approach such problems, one technique is to rearrange the equation to one side to resemble \(y <\) or \(y >\) a quadratic expression. This allows us to leverage the visual power of graphing to identify the regions that satisfy the inequality. Understanding that the inequality sign \(>\) indicates values above the graph, while \(<\) signifies values below the graph, is crucial for interpreting the solution.

In the realm of algebra, parabola graphing is a fundamental skill when dealing with quadratic functions. Every quadratic function graphs into a parabola, a U-shaped curve that may open upwards or downwards depending on the coefficient of the \(x^{2}\) term.

An upward opening parabola has the general formula \(y = ax^{2} + bx + c\), with \(a > 0\), whereas a downward opening parabola reverses situation with \(a < 0\). The most distinctive point on a parabola is its vertex, which represents the maximum or minimum point of the curve. For an upward opening parabola, as in our example \(y = 2x^{2} - 3\), the vertex is a minimum point. Graphing a parabola involves plotting sufficient points to outline its shape or using a graphing utility for accuracy and to visualize where solutions to an inequality may lie.

Inequality solutions refer to the set of values that fulfill the requirements of a given inequality. When graphing inequalities involving a quadratic, as seen with \(y < 2x^{2} - 3\), the solution is a region rather than a single point or line.

This region is visually represented on a graph by shading. For \(y < ax^{2} + bx + c\), the solution is all the points below the parabola. Conversely, if the inequality were \(y > ax^{2} + bx + c\), the solution would be shaded above. It's important to note that the inequality \(y <\) or \(y >\) does not include the line \(y = ax^{2} + bx + c\), unless indicated with a 'less than or equal to' \(\leq\) or 'greater than or equal to' \(\geq\) sign.

Lastly, in interpreting solutions for inequalities, keep in mind that with quadratic inequalities the solutions often come in intervals, which can be represented either graphically or in interval notation.