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Chapter 10: Problem 35

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

Short Answer

Expert verified
Vertices: (2, 4), (-2, 4). The solution set is bounded.

Step by step solution

01

Graph the Boundary of the First Inequality

The inequality \(x^2 - y \leq 0\) can be rewritten as \(y \geq x^2\). This represents the region above (or on) the parabola \(y = x^2\). To graph this, draw the parabola \(y = x^2\) and shade the region above it.
02

Graph the Boundary of the Second Inequality

The inequality \(2x^2 + y \leq 12\) can be rewritten as \(y \leq 12 - 2x^2\). This represents the region below (or on) the parabola \(y = 12 - 2x^2\). To graph this, draw the parabola \(y = 12 - 2x^2\) and shade the region below it.
03

Find the Intersection of the Parabolas

To find where the parabolas intersect, solve the equations \(y = x^2\) and \(y = 12 - 2x^2\) simultaneously. Setting \(x^2 = 12 - 2x^2\), we get: \[3x^2 = 12\] \[x^2 = 4\] \[x = \pm 2\]. When \(x = 2\), \(y = 4\); and when \(x = -2\), \(y = 4\). Therefore, the parabolas intersect at (2, 4) and (-2, 4).
04

Identify and Label the Vertices of the Solution Region

The vertices of the solution region come from intersections of the boundaries and boundary lines themselves. Besides the intersection points found, check if the parabolas' end behaviors or other borders might result in additional vertices enclosed by both conditions. Here, parabolas connect only at (2, 4) and (-2, 4) forming necessary vertices.
05

Determine if the Solution Set is Bounded

The solution set may be bounded if it is enclosed within the graph and doesn't extend infinitely in any direction. As both parabolas extend infinitely, but in opposite directions, the overlapping/shaded solution region hosted between them forms an enclosed shape, making the solution set bounded. Trace the vertices carrier shapes to conclude.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Parabolas in Inequalities
When working with systems of inequalities, understanding the role of parabolas is important. A parabola is a symmetrical valley or arch-shaped curve on a graph. In algebra, it's commonly expressed in quadratic form, such as \(y = ax^2 + bx + c\). Each parabola has a vertex, which is its peak or lowest point.
For this system, we have two parabolas: \(y = x^2\) and \(y = 12 - 2x^2\). Each describes one side of the inequality and defines a region on the graph. To find where the inequalities apply, we not only plot these curves but also shade the relevant regions. The solution lies in their overlap.
Parabolas in the inequalities matter as they form boundaries. For \(y \geq x^2\), it describes the area above \(y = x^2\). Meanwhile, \(y \leq 12 - 2x^2\) describes below the second parabola. Identifying how these regions overlap is crucial in solving the system.
Finding the Solution Set
The solution set is the part of the graph that satisfies all the inequalities in the system. To determine the solution set, we need to consider where the shaded regions overlap.
When the boundary lines were graphed, the curve \(y = x^2\) had its region shaded above it. The curve \(y = 12 - 2x^2\) required shading below it. Where these shades intersect is the solution set.
In step 3, solving \(x^2 = 12 - 2x^2\) gave us intersection points (2, 4) and (-2, 4). These points are important as they define the corners of the overlap, or vertices. Shaded areas around these include all points that satisfy both inequalities, completing the solution set.
Deciding if the Solution Set is Bounded
A solution set is bounded if it's confined within a specific area on a graph and does not extend infinitely. With systems of inequalities, determining boundedness helps us understand if the region is finite.
In our exercise, both parabola parabolas extend infinitely. However, when combined, they form a contained region that doesn鈥檛 stretch indefinitely in any direction. This makes the solution set bounded.
The parabolas meet at specific vertices, (2, 4) and (-2, 4). These intersections, along with the direction of shading, enclose the solution set. By examining these factors, we conclude that the solution of this system of inequalities resides within a bounded region.

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Most popular questions from this chapter

Nutrition A doctor recommends that a patient take \(50 \mathrm{mg}\) each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home the patient finds three brands of vitamin pills. The amounts of the relevant vitamins per pill are given in the table. How many pills of each type should he take every day to get \(50 \mathrm{mg}\) of each vitamin? $$\begin{array}{|l|c|c|c|} \hline & \text { VitaMax } & \text { Vitron } & \text { VitaPlus } \\ \hline \text { Niacin (mg) } & 5 & 10 & 15 \\ \text { Riboflavin (mg) } & 15 & 20 & 0 \\ \text { Thiamin (mg) } & 10 & 10 & 10 \\ \hline \end{array}$$

An encyclopedia saleswoman works for a company that offers three different grades of bindings for its encyclopedias: standard, deluxe, and leather. For each set that she sells, she earns a commission based on the set's binding grade. One week she sells one standard, one deluxe, and two leather sets and makes \(\$ 675\) in commission. The next week she sells two standard, one deluxe, and one leather set for a \(\$ 600\) commission. The third week she sells one standard, two deluxe, and one leather set, earning \(\$ 625\) in commission. (a) Let \(x, y,\) and \(z\) represent the commission she earns on standard, deluxe, and leather sets, respectively. Translate the given information into a system of equations in \(x, y\) and \(z\) (b) Express the system of equations you found in part (a) as a matrix equation of the form \(A X=B\). (c) Find the inverse of the coefficient matrix \(A\) and use it to solve the matrix equation in part (b). How much commission does the saleswoman earn on a set of encyclopedias in each grade of binding?

Use a graphing calculator to graph the solution of the system of inequalities. Find the coordinates of all vertices, rounded to one decimal place. $$\left\\{\begin{array}{l} y \geq x-3 \\ y \geq-2 x+6 \\ y \leq 8 \end{array}\right.$$

Sketch the triangle with the given vertices, and use a determinant to find its area. $$(1,0),(3,5),(-2,2)$$

Find the inverse of the matrix. For what value(s) of \(x\) if any, does the matrix have no inverse? $$\left[\begin{array}{ccc} 1 & e^{x} & 0 \\ e^{x} & -e^{2 x} & 0 \\ 0 & 0 & 2 \end{array}\right]$$
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