Problem 16 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1 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 \(x^{2}+y^{2}=r^{2}\).

Step 2 Draw a Graph

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

Step 3 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.

Step 4 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.

Key Concepts

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.