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A quadratic inequality is an inequality involving a quadratic relation in one or two variables. For example,

y≥3x2−2x−4,

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.Graphing a quadratic inequality is similar to graphing a quadratic function, but instead of a parabola, the graph is an entire region.

To graph the quadratic inequalityy<x2+2x−1,

draw the boundary, determine if the solution set lies inside or outside the parabola, and shade the region that contains the solution set.
Graph the boundary

y=x2+2x−1.

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 y-intercept, and the symmetry inherent to a parabola.
Test a point

Shade the appropriate region

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 (0,0).

The region containing (0,0) is inside the parabola. Since (0,0) 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.

$(-3,-4),(1,2),and(3,7)$

Show Solution

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,$ (-3,-4)(1,2) is not a solution,butis a solution. $

The point (3,7) lies on the boundary. However, since the curve is dashed, points on the boundary are not included in the solution set. Therefore, the point $ (3,7) is not a solution. $

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