/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Systems of Equations and Inequal... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 16

Systems of Equations and Inequalities. $$x^{2}+y^{2}>36$$

Short Answer

Expert verified
The solution to the inequality \(x^{2}+y^{2} > 36\) is the area outside the circle centered at the origin with a radius of 6, excluding the circle itself.

Step by step solution

01

Identify the inequality

Identify the inequality equation as \(x^{2}+y^{2} > 36\) which symbolizes an area outside a circle centered at (0,0) and with a radius of 6 (since \(\sqrt{36}=6\)). This comes from the standard form of the circle equation which is \(x^{2}+y^{2}=r^{2}\).
02

Draw a Graph

Proceed to draw a graph. Start by drawing a circle centered at (0,0) with a radius of 6.
03

Shade the Area

Considering that the inequality is greater than (>), and not equal to (=), we should shade the area outside the circle to present the solution. But, remember not to include the circle in shading as it is not part of the solution.
04

Interpret the Graph

The graphed region stands for the solutions to the inequality. Every point (x,y) that is located in the shaded area of the graph is a solution to the inequality. Therefore a graph representation properly outlines the solution region, for the inequality.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Circle Equation
Understanding the circle equation is the first step in tackling the inequality problem provided. A circle in a coordinate plane can be described using the equation \[x^2 + y^2 = r^2\]where \(r\) is the radius of the circle, and the circle is centered at the origin (0,0) if not shifted. This equation defines all the points that are exactly \(r\) distance away from the center. For example, if \(r = 6\), the equation becomes \(x^2 + y^2 = 36\). This means every point (x,y) on the circle is 6 units away from the center at (0,0).
So, the equation \(x^2 + y^2 = 36\) describes a perfect circle centered at the origin with a radius of 6 units.
Inequality Graphing
Graphing inequalities involving circles requires a bit more than simply drawing a circle. With the inequality \(x^2 + y^2 > 36\), we focus on all the points outside of the circle defined by \(x^2 + y^2 = 36\). Here's what you do:
  • Draw the circle as per \(x^2 + y^2 = 36\).
  • Use a dashed line for the circle instead of a solid one. This indicates that the points on the circle itself are not part of the solution set, aligning with the \(>\) (greater than) sign in the inequality.
  • Shade the entire region outside of the dashed circle. This shaded area represents all (x,y) points where \(x^2 + y^2 > 36\).
This graphing method is key to visually solving such inequalities.
Solution Region
The solution region for the inequality \(x^2 + y^2 > 36\) is the area outside the shadow of the circle drawn on the Cartesian plane. To understand it clearly:
  • Consider all points (x,y) that do not satisfy the equality \(x^2 + y^2 = 36\).
  • These points are every (x,y) that lies outside the given circle.
  • Such points have distances from the center greater than 6 units.
To emphasize, for any point outside the circle, plotting it on the graph should satisfy the inequality, meaning when substituted into \(x^2 + y^2\), the result is greater than 36.
Graph Interpretation
Interpreting a graph of this inequality involves understanding what it visually represents:
  • The circle itself, drawn with a dashed line, is just a boundary that helps identify the division between points that do not satisfy the inequality, and those that do.
  • The shaded area outside suggests infinite potential solutions that make \(x^2 + y^2 > 36\) true.
  • Any point located in this shaded area, when checked, will have an \(x^2 + y^2\) value greater than 36.
This interpretation helps pinpoint possible solutions and assists in visualizing where on the plane these solutions lie, facilitating better understanding and problem-solving.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The table shows the price of a gallon of unleaded premium gasoline. For each price, the table lists the number of gallons per day that a gas station sells and the number of gallons per day that can be supplied. $$\begin{array}{lll}{\text { Price per }} & {\text { Gallons Demanded }} & {\text { Gallons Supplied }} \\ {\text { Gallon }} & {\text { per Day }} & {\text { per Day }} \\ {\$ 3.20} & {1400} & {200} \\ {\$ 3.60} & {1200} & {600} \\ {\$ 4.40} & {800} & {1400} \\ {\$ 4.80} & {600} & {1800}\end{array}$$ The data in the table are described by the following demand and supply models: Demand Model \(\quad\) Supply Model \(p=-0.002 x+6 \quad p=0.001 x+3\) a. Solve the system and find the equilibrium quantity and the equilibrium price for a gallon of unleaded premium gasoline. b. Use your answer from part (a) to complete this statement: If unleaded premium gasoline is sold for _____ per gallon, there will be a demand for ______ gallons per day and ______ gallons will be supplied per day.

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$ \begin{aligned} &z=4 x+y\\\ &\left\\{\begin{array}{l} {x \geq 0, y \geq 0} \\ {2 x+3 y \leq 12} \\ {x+y \geq 3} \end{array}\right. \end{aligned} $$

Given \(f(x)=6 x+5\) and \(g(x)=x^{2}-3 x+2,\) find each of the following: a. \((f \circ g)(x)\) b. \((g \cdot f)(x)\) c. \((f \circ g)(-1)\)

What kinds of problems are solved using the linear programming method?

will help you prepare for the material covered in the next section. Graph \(x-y=3\) and \((x-2)^{2}+(y+3)^{2}=4\) in the same rectangular coordinate system. What are the two intersection points? Show that each of these ordered pairs satisfies both equations.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.