Question

Graph the solution set of the system of inequalities. $$\left\\{\begin{array}{rr}x^{2}+y \leq & 6 \\ x \geq & -1 \\ y \geq & 0\end{array}\right.$$

Short answer

To solve the system of inequalities, they must sketch each inequality on the same graph and identify the region where all the inequalities are satisfied. The solution set will be the intersection of all these solution sets.

Step-by-step solution

Sketch the Inequality \(x^{2}+y \leq 6\)

This inequality represents a parabola opening upwards which is situated below the line \(y = 6 - x^{2}\). So, the solution of this inequality will be the interior part of the parabola along with the edge.

Sketch the Inequality \(x \geq -1\)

This inequality forms a vertical line at \(x=-1\) and the solution for it will be all points to the right side of this line including the line itself.

Sketch the Inequality \(y \geq 0\)

This inequality makes a horizontal line at \(y = 0\) and the solution set for this inequality will be all points above and on this line.

Identify the Solution Region of the System of Inequalities

The solution set of the system of inequalities will be the intersection of solutions from each inequality. This is the area of graph where all the shaded regions overlap.

Key concepts

Parabola Graphing

In order to graph a parabola when solving systems of inequalities, it's helpful to understand the basic structure of a parabola. A parabola that involves a quadratic expression with an \(x^2\) term, like \(x^2 + y \leq 6\), forms a curve with a specific shape, typically either opening up or down.

The line defining the parabola can be found by equating the inequality to equality, that is introducing \(y = 6 - x^2\). Here, the line \(y = 6 - x^2\) serves as the boundary of the solution region. Since we're dealing with an inequality \((\leq)\), the parabola contains the area inside and on the boundary line.

• The vertex of the parabola is critical and can be found at the highest or lowest point of the graph. In our case, the vertex is located at the maximum point, (0,6).
• The arms of the parabola move downwards from the vertex.

Making sure to shade the correct region underneath the line is essential in determining the solution set of the inequality.

Inequality Solutions

When solving inequalities graphically, understanding how each inequality functions individually prepares the ground. For the inequality \(x \geq -1\), it means that our graph should include the line \(x = -1\) and all points to the right of it. This is depicted by shading on the right side of this vertical line.

For \(y \geq 0\), you want to locate the line \(y = 0\), which is simply the x-axis, and include every point above and on this line. Such straightforward inequalities often form a large part of the solution space.

• Each line forms a boundary in the plane.
• By using dashed or solid lines, students represent the difference between strict and non-strict inequalities.

Solid lines, as in our example, indicate that boundary lines are included in the solution set.

Graphical Solution Methods

The graphical method for solving systems of inequalities involves overlaying multiple solution sets derived from individual inequalities. The system's solution is where these sets overlap, marking the feasible region within the defined boundaries.

Start by graphing each inequality separately. Pay careful attention to the type of boundary line you need to use—solid for inclusive inequalities like \( \leq \) or \( \geq \).

• Validate the drawing of each inequality graph with test points if necessary.
• Ensure each shaded area correctly reflects its associated inequality.
• The solution region is the shared space where all individual solutions layer together.

Double-checking this overlapping region ensures accurate identification of the solution set. The graphical method provides a visual confirmation and is instrumental when systems become more complex. It makes seeing the interactions of each inequality simpler.