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Concept

A quadratic inequality is an inequality involving a quadratic relation in one or two variables. For example, $y\geq 3x^2-2x-4,$

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.
Method

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 inequality $y< x^2+2x-1,$ draw the boundary, determine if the solution set lies inside or outside the parabola, and shade the region that contains the solution set.

### 1

Graph the boundary

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 $y=x^2+2x-1.$ If the symbol is $<$ or $>,$ the boundary is dashed, and solid if the symbol is $\leq$ or $\geq.$ Here, it will be dashed. The boundary can be graphed using the vertex, the $y$-intercept, and the symmetry inherent to a parabola.

### 2

Test a point
The solution set of the inequality either lies inside or outside the parabola. To determine which, substitute an arbitrarily chosen test point (not on the boundary) into the inequality to find if it is a solution. Using $(0,0)$ is preferable since the calculations usually become relatively simple.
$y< x^2+2x-1$
${\color{#009600}{0}}\stackrel{?}{<} {\color{#0000FF}{0}}^2+2\cdot {\color{#0000FF}{0}}-1$
$0\nless \text{-} 1$
Since the test point made a false statement, it is not a solution to the inequality.

### 3

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.

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Exercise

Use the graph to determine if the following points are solutions to the corresponding inequality graphed in the coordinate system. Justify your answer. $(\text{-} 3, \text{-} 4), \quad (1,2), \quad \text{ and } \quad (3,7)$

Show Solution
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, \begin{aligned} &(\text{-} 3,\text{-} 4) && \text{is not a solution, but } \\ &(1,2) && \text{is a solution.} \end{aligned} 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 \begin{aligned} &(3,7) && \text{is not a solution.} \end{aligned}