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Solving Quadratic Systems

Solving Quadratic Systems 1.7 - Solution

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We want to solve the given system of inequalities by graphing. Note that both inequalities of the system are quadratic inequalities. Let's graph each of them, one at a time.

Inequality (I)

We can write the boundary curve for Inequality (I) by replacing the greater than sign with an equals sign. Then, we can identify and Knowing that and we can find the vertex. To do so, we will need to think of as a function of Let's substitute the values of and in the formula for the coordinate of the vertex.
The coordinate of the vertex is Now, let's find the coordinate by substituting for into the quadratic equation for the boundary line.
Simplify right-hand side
The vertex is With this, we know that the axis of symmetry of the parabola is the vertical line Next, let's find two more points on the curve, one on each side of the axis of symmetry.

The points and are on the parabola. Let's plot the points and connect them with a smooth curve.

Now that we have the boundary curve, we need to determine which region to shade. To do so, we will use as a test point. If the point satisfies the inequality, we will shade the region that contains the point. If not, we will shade the opposite region.
Simplify right-hand side
Since the substitution did not produce a true statement, we will shade the region which does not contain the point Because we have a strict inequality, the boundary curve will be dashed.

Inequality (II)

Once again, we can write the boundary curve by replacing the less than sign with an equals sign. We can draw the second parabola following the same procedure as with the first.

Vertex Axis of Symmetry Two Points

Any point on the plane can be used to determine the region we should shade. Because this inequality is strict, the curve will be dashed.

Final Solution Set

The solution set is the overlapping region.