/*! 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 20 Graph each inequality. \(4 y^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 20

Graph each inequality. \(4 y^{2} \leq 36-9 x^{2}\)

Short Answer

Expert verified
The inequality represents an ellipse centered at (0,0) with semi-major axis of 3 units along y-axis and semi-minor axis of 2 units along x-axis. Shade the interior of this ellipse.

Step by step solution

01

Rewrite the Inequality in Standard Form

Start by rewriting the given inequality \(4 y^{2} \leq 36-9 x^{2}\) in a standard form. First, move all terms to one side to get \(4 y^{2} + 9 x^{2} \leq 36\).
02

Simplify the Inequality

Divide both sides of the inequality by 36 to simplify. \(\frac{4 y^{2}}{36} + \frac{9 x^{2}}{36} \leq 1\) simplifies to \(\frac{y^{2}}{9} + \frac{x^{2}}{4} \leq 1\).
03

Identify the Conic Section

Recognize that \(\frac{y^{2}}{9} + \frac{x^{2}}{4} \leq 1\) represents an ellipse centered at the origin \( (0,0) \) with semi-major axis y (length 3 units) and semi-minor axis x (length 2 units).
04

Graph the Ellipse

To graph the ellipse, plot the center at (0,0). Then plot points at (0,3), (0,-3), (2,0), and (-2,0). Draw the ellipse passing through these points.
05

Shade the Interior Region

Since the inequality is \(\frac{y^{2}}{9} + \frac{x^{2}}{4} \leq 1\), shade the interior region of the ellipse to indicate all the points (x, y) that satisfy 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.

headline of the respective core concept
First concept involves the Standard Form. Getting inequalities into this form makes solving easier.
Start by moving all terms to one side of the inequality.
This action helps form a comparison against zero, vital for observing the inequality's relationships. For example, with our inequality, transfer all terms to one side to get:
\[ 4 y^{2} + 9 x^{2} \- 36 \ leq 0. \] This format allows identifying, simplifying and graphing conic sections like ellipses.
headline of the respective core concept
Simplifying inequalities is crucial for clarity and ease in graphing.
Simplify your inequality by dividing or factoring common terms.
With our example, divide by 36: \[ \frac{4 y^{2}}{36} + \frac{9 x^{2}}{36} \leq 1.\] Notice how it simplifies to: \[ \frac{y^{2}}{9} + \frac{x^{2}}{4} \leq 1. \] This provides a more recognizable form.
headline of the respective core concept
Understanding Conic Sections is important in graphing. Conic sections are curves obtained by intersecting a cone with a plane.
They include circles, ellipses, parabolas, and hyperbolas.
Our simplified inequality, \[ \frac{y^{2}}{9} + \frac{x^{2}}{4} \leq 1 \] represents an ellipse, which is a specific type of conic section.
headline of the respective core concept
Ellipses are conic sections that look like elongated circles.
The standard form for an ellipse, centered at the origin, is: \[ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \] For our example, \[ \frac{y^{2}}{9} + \frac{x^{2}}{4} \leq 1, \] with 'a' = 3 and 'b' = 2.
This means semi-major axis (3 units along y-axis) and semi-minor (2 units along x-axis).
headline of the respective core concept
Last is the Interior Region. After graphing the ellipse, identify the interior.
Our inequality, \[ \frac{y^{2}}{9} + \frac{x^{2}}{4} \leq 1 \] tells us to consider points inside the ellipse, not just on boundary.
Shade this interior to show the area where the inequality holds true.
This visual helps in understanding solutions to the inequality easily and correctly.

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

Graph each inequality. \(x^{2}>4-y^{2}\)

Graph each system of inequalities. \(5 x-3 y \leq 15\) \(4 x+y \geq 4\)

Write a system of inequalities for the indicated region. Quadrant I, including the \(x\) - and \(y\) -axes

Graph each ellipse. $$\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$$

Graph each ellipse. $$\frac{y^{2}}{25}=1-\frac{x^{2}}{49}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.