A quadratic inequality is an inequality involving a quadratic relation in one or two variables. For example,is a quadratic inequaliy. Similar to linear inequalities, the solution set to a quadratic inequality is an entire region of the coordinate plane. However, instead of the boundary being a line, it is a parabola.
The boundary of the inequality is the parabola corresponding to the equation produced if the inequality symbol is replaced with an equals sign. In this case, this is the parabola If the symbol is or the boundary is dashed, and solid if the symbol is or Here, it will be dashed. The boundary can be graphed using the vertex, the -intercept, and the symmetry inherent to a parabola.
If the test point is a solution to the inequality, the region in which it lies contains the entire solution set. If not, the other region represents the solution set. Here, the test point is
The region containing is inside the parabola. Since is not a solution, the region outside the parabola containts the solution set.
Use the graph to determine if the following points are solutions to the corresponding inequality graphed in the coordinate system. Justify your answer.
The graph shows the solution set to the inequality. Let us begin by marking the three points on the coordinate plane.
A point that lies within the shaded region is a solution to the inequality, while a point that lies outside is not. Therefore, The point lies on the boundary. However, since the curve is dashed, points on the boundary are not included in the solution set. Therefore, the point