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Chapter 6: Problem 63

Graph the solution set of each system of inequalities by hand. Concept Check \(\quad\) Which one of the choices that follow is a description of the solution set of the following system? $$ x^{2}+y^{2} < 36 $$ \(y < x\) A. All points outside the circle \(x^{2}+y^{2}=36\) and above the line \(y=x\) B. All points outside the circle \(x^{2}+y^{2}=36\) and below the line \(y=x\) C. All points inside the circle \(x^{2}+y^{2}=36\) and above the line \(y=x\) D. All points inside the circle \(x^{2}+y^{2}=36\) and below the line \(y=x\)

Short Answer

Expert verified
Answer: D. All points inside the circle and below the line.

Step by step solution

01

Understand the Circle Inequality

The inequality \(x^2 + y^2 < 36\) represents the set of all points inside the circle with center at the origin \((0, 0)\) and radius 6. This includes all points such that the sum of the squares of their \(x\) and \(y\) coordinates is less than 36.
02

Understand the Line Inequality

The inequality \(y < x\) represents the region below the line \(y = x\). This means any point \((x, y)\) where the y-value is less than the x-value falls below this diagonal line.
03

Determine the Overlapping Region

To find the solution set for the system of inequalities, determine the region where both conditions are satisfied. This means identifying points that are both inside the circle \(x^2 + y^2 < 36\) and below the line \(y = x\).
04

Identify the Correct Description

From the analysis above, we need points that are inside the circle and below the line \(y = x\). The correct description matching this region is D: All points inside the circle \(x^2 + y^2=36\) and below the line \(y = x\).

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Key Concepts

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

Graphing Inequalities
Graphing inequalities involves representing regions on a coordinate plane that satisfy given conditions. By graphing, you can visually understand where certain equations hold true. For each inequality, you'll identify and shade the right area.

When graphing:
  • Start by plotting the equation as if it were an equality (e.g., for \(y < x\), plot \(y = x\).
  • Choose a test point not on the line to see which side satisfies the inequality.
  • Shade the region that includes all points satisfying the inequality.
This visualization helps to see where two or more conditions overlap, finding the solution to a system of inequalities.
Circle Inequality
The circle inequality \(x^2 + y^2 < 36\) describes a set of points within a circle centered at \( (0, 0) \) with a radius of 6. This means all the points whose distance from the origin is less than 6 will satisfy this inequality.

Understanding this concept involves knowing:
  • Standard circle equation: \(x^2 + y^2 = r^2\).
  • In \(x^2 + y^2 < 36\), \(r = 6\), and you're looking for points that lie inside.
This inequality draws an open circle, which means it doesn't include the boundary circle itself.
Line Inequality
The inequality \(y < x\) represents all points that lie below the diagonal line \(y = x\). This line acts as a boundary separating where points are either greater or less than each other.

Key aspects to remember:
  • The line \(y = x\) has a slope of 1, passing through the origin.
  • For \(y < x\), you鈥檙e interested in the region where the \(y\)-value is less than the \(x\)-value.
Shading the area below this line on the graph visually shows all solutions to this inequality.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &y \geq 3^{x}\\\ &y \geq 2 \end{aligned}$$

In certain parts of the Rocky Mountains, deer are the main food source for mountain lions. When the deer population \(d\) is large, the mountain lions ( \(m\) ) thrive. However, a large mountain lion population drives down the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$\left[\begin{array}{c} m_{n+1} \\ d_{n+1} \end{array}\right]=\left[\begin{array}{cc} 0.51 & 0.4 \\ -0.05 & 1.05 \end{array}\right]\left[\begin{array}{l} m_{n} \\ d_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 year? 2 years? (c) Consider part (b), but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of \(1 \$ 6\).

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}y &=5^{x} \\\x y &=1\end{aligned}$$

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A line and a parabola; no points.

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rr} \frac{3}{20} & \frac{1}{4} \\ -\frac{1}{20} & \frac{1}{4} \end{array}\right]$$
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