Problem 68 - Algebra 2

Question

Solve each system of inequalities by graphing. $$ \begin{array}{l}{9 x^{2}+y^{2}<81} \\ {x^{2}+y^{2} \geq 16}\end{array} $$

Step-by-Step Solution

Answer

The solution is the region outside the circle and inside the ellipse.

Step 1: Understand Each Inequality

The system of inequalities consists of two parts:1) \(9x^2 + y^2 < 81\) and 2) \(x^2 + y^2 \geq 16\). The first inequality represents the interior of an ellipse and the second represents the area outside of or on the boundary of a circle.

Step 2: Rewrite the Inequalities for Comparison

First, simplify the inequalities:1) For \(9x^2 + y^2 < 81\), divide by 81 to get \(\frac{x^2}{9} + \frac{y^2}{81} < 1\).2) For \(x^2 + y^2 \geq 16\), it is already in the standard form of a circle's inequality.

Step 3: Graph the Ellipse

Draw the ellipse defined by \(\frac{x^2}{9} + \frac{y^2}{81} = 1\). Its center is at the origin (0,0), with semi-major axis length 9 (vertical) and semi-minor axis length 3 (horizontal). The region inside this ellipse is shaded to represent where \(9x^2 + y^2 < 81\).

Step 4: Graph the Circle

Draw the circle given by \(x^2 + y^2 = 16\). Its center is at the origin with a radius of 4. Shade the region that is not inside this circle to show \(x^2 + y^2 \geq 16\) which represents everything outside this circle and including the boundary.

Step 5: Identify the Overlap

The solution to the system of inequalities is the region where the shaded area from the ellipse (interior) and the shaded area from the circle (exterior) overlap. This is the area outside or on the boundary of the circle and inside the ellipse.

Key Concepts

Ellipse Inequality

An ellipse inequality describes a region related to an ellipse's geometry. For instance, the inequality \( 9x^2 + y^2 < 81 \), when simplified to \( \frac{x^2}{9} + \frac{y^2}{81} < 1 \), outlines the inside of an ellipse centered at the origin (0,0).

This specific ellipse has unique dimensions:

• Its semi-major axis is vertical with a length of 9, making it more stretched along the y-axis.
• The semi-minor axis is horizontal with a length of 3, giving it a compressed appearance horizontally.

To graph this inequality, imagine the boundary of the ellipse as \( \frac{x^2}{9} + \frac{y^2}{81} = 1 \). The region fulfilling the inequality is within this boundary, forming a closed shape that resembles a squashed circle.

Circle Inequality

Circle inequalities represent regions involving circles. In our system, the inequality \( x^2 + y^2 \geq 16 \) describes regions outside or on the perimeter of a circle. This circle is centered at the origin, (0,0), and has a radius of 4.

• The "\( \geq \)" symbol indicates that the region includes the circle's boundary and everything outside it.
• This circle's radius of 4 means any point within 4 units of the center is excluded from the solution region.

Graphing the circle inequality involves drawing the circle \( x^2 + y^2 = 16 \) with a solid line to show the boundary is part of the solution. The area that satisfies the inequality is shaded outside this circle.

Graphing Inequalities

Graphing inequalities involves visually representing solutions on a coordinate plane. This starts by understanding each inequality's sign:

• For "<" or ">",

Graphing Conic Sections

Conic sections include ellipses and circles, which both appear in our system of inequalities. When graphing conic sections, their equations help determine the shapes' key features:

• Circles, like \( (x-a)^2 + (y-b)^2 = r^2 \), are defined by their center (a, b) and radius r.
• Ellipses are characterized by their semi-major and semi-minor axes lengths and orientation, as seen in \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \).

The process of graphing these sections involves sketching their precise geometric boundaries. By accurately sketching and shading the appropriate regions, a clear visual representation of the inequalities' solutions is achieved. Understanding these conic sections aids in identifying intersections and overlaps in complex systems, ensuring solutions like those inside an ellipse and outside a circle in our problem.