Graphing Quadratic Inequalities

To graph the quadratic inequality, we will follow three steps.

1. Graph the related quadratic function.
2. Test a point not on the parabola.
3. Shade accordingly. If the point satisfies the inequality, we shade the region that contains the point. If not, we shade the opposite region.

Step $1$

Let's draw the graph of the related function, which is $y={\left(x-\frac{1}{2}\right)}^2+\frac{5}{2}.$

Step $2$

Next, let's determine which region to shade by testing a point. For simplicity, we will use $(0,0)$ as our test point. Let's see if it satisfies the given inequality.

Step $3$

Since $(0,0)$ produced a true statement, we will shade the region that contains the point. Also, note that the inequality is not strict. Therefore, the parabola will be solid.