Problem 29 - Precalculus: Mathematical for Calculus

Question

Graph the solution of the system of inequalities. Find the coordinates of all vertices, and determine whether the solution set is bounded. $$\left\\{\begin{array}{c}x^{2}-y \leq 0 \\\2 x^{2}+y \leq 12\end{array}\right.$$

Step-by-Step Solution

Answer

The vertices are \((-2, 4)\) and \((2, 4)\), and the solution set is bounded.

Step 1: Rewrite inequalities

We begin by rewriting each inequality in a form that is easier to work with for graphing. The first inequality is given as \(x^2 - y \leq 0\), which can be rewritten as \(y \geq x^2\). The second inequality is \(2x^2 + y \leq 12\), which can be rewritten as \(y \leq 12 - 2x^2\).

Step 2: Graphing the equations

Now, graph the curves corresponding to the equations \(y = x^2\) and \(y = 12 - 2x^2\) on the same set of axes. These curves serve as boundaries of the regions defined by the inequalities.

Step 3: Identify the solution region

The solution region is the area where the solution of both inequalities intersect. It's the region that lies above the parabola \(y = x^2\) and below the parabola \(y = 12 - 2x^2\).

Step 4: Finding intersection points

To determine the vertices of the solution region, we need the intersection points of the boundary equations. Set \(x^2 = 12 - 2x^2\) and solve for \(x\). This gives \(3x^2 = 12\), which simplifies to \(x^2 = 4\). Therefore, \(x = \pm 2\). Substitute back to find \(y\): when \(x = 2\), \(y = 4\), and when \(x = -2\), \(y = 4\). Thus, the vertices are \((2, 4)\) and \((-2, 4)\).

Step 5: Determine if the solution set is bounded

A region is bounded if it is enclosed within finite limits. Since the region above \(y = x^2\) and below \(y = 12 - 2x^2\) forms a closed area around the vertices \((-2,4)\), \((2,4)\), and covering part of the lines, it is bounded.

Key Concepts

Solution Region

When solving systems of inequalities, the "solution region" is the space on a graph where all of the inequalities in the system are true at the same time. Imagine this as the area where all shading overlaps when each inequality is graphed individually.
For the inequalities provided:

• Rewriting the first inequality, we get \( y \geq x^2 \). This represents all points above, or on, the parabola \( y = x^2 \).
• The second inequality becomes \( y \leq 12 - 2x^2 \). This covers the points below, or on, the parabola \( y = 12 - 2x^2 \).

The solution region is the intersection of these two shaded areas, which means it is above the parabola \( y = x^2 \) and below \( y = 12 - 2x^2 \). Therefore, the solution region meets both conditions set by the inequalities, creating an area that satisfies both equations simultaneously.

Intersection Points

Intersection points are where the solutions to equations meet, and they are crucial for outlining solution regions in inequalities. These points help us identify the vertices of the solution region.
For our system of inequalities, finding the intersection points involves solving the equations \( x^2 = 12 - 2x^2 \) for \( x \). Simplifying this equation gives \( 3x^2 = 12 \), which results in the solutions \( x = \pm 2 \).
By substituting \( x = 2 \) and \( x = -2 \) into either of the original equations, we find the corresponding \( y \)-values. Both substitutions yield \( y = 4 \). Therefore, the intersection points, or vertices, of the solution region are \( (2, 4) \) and \( (-2, 4) \). These points define the critical bounds of our solution region, showing where the two graph lines intersect.

Bounded Region

A region in the context of graphing inequalities is described as "bounded" if it is enclosed and finite. Visualizing this involves imagining a fence that completely surrounds the solution area on a graph.
The solution region formed by the parabolas \( y = x^2 \) and \( y = 12 - 2x^2 \) is a closed curve above and below the intersection points and extends sideways.

• It is closed along the boundaries where \( x = 2 \) and \( x = -2 \), the vertical lines that join the two parabolas at the vertices \( (2, 4) \) and \( (-2, 4) \).
• The whole area inside these bounds, above the parabola \( y = x^2 \) and below \( y = 12 - 2x^2 \), forms the bounded region.

Hence, this solution region is confined within finite limits in all directions, confirming it as a bounded region. Understanding bounded regions helps in determining the scope and limits of solutions in the coordinate plane.