Problem 16 - College Algebra

Question

Graph each inequality. $$ x^{2}+y^{2}>36 $$

Step-by-Step Solution

Answer

The drawing must have a circle centered at (0,0) with radius 6 units. The region outside the circle is shaded and arrows pointing outwards from the circle are drawn to represent the set of solutions for the given inequality.

Step 1: Draw the Circle

Firstly, draw a circle centered at the origin (0,0) with a radius of 6 units. This is because the equation of a circle with radius r and center (h, k) is \( (x-h)^{2} + (y-k)^{2} = r^{2} \). Here, h = k = 0 and r = 6, so the equation becomes \(x^{2} + y^{2} = 36\). Notice that the inequality sign is replaced by equal sign, because we are drawing a circle boundary.

Step 2: Shade the Required Region

Since the inequality is \(x^{2} + y^{2} > 36\), all the points satisfying this inequality lie outside the circle. We shade the region outside the circle to represent all such points. Ensure not to shade the circle itself, since the inequality is a strict inequality (>).

Step 3: Indicate the Direction

Sometimes, it is helpful to indicate the direction of the inequality with arrows. Since the inequality is \(x^{2} + y^{2} > 36\), we can draw radial arrows pointing outward from the circle to represent that all points outside of the circle are solutions.

Key Concepts

Circle Equations

A circle is a set of points equidistant from a fixed center point. In the equation form \( (x-h)^2 + (y-k)^2 = r^2 \), the center is \((h, k)\) and \(r\) is the radius. For an origin-centered circle, this simplifies to \( x^2 + y^2 = r^2 \). This is because both \(h\) and \(k\) are zero.

Circle equations are fundamental in graphing because they define the boundary of the circle. By understanding this setup, you can identify what part of the plane relates to the circle.

Strict Inequalities

A strict inequality does not include the boundary value. It’s shown by the symbols \(>\) or \(<\). When you graph a strict inequality like \(x^2 + y^2 > 36\), the boundary circle itself isn't included.

• To represent this, you typically draw a dotted line for the circle.
• This shows that the points on the circle are not part of the solution.

This is crucial for accurately showing which regions satisfy the inequality.

Shading Regions

Shading helps to visualize which part of the graph satisfies the inequality. For \(x^2 + y^2 > 36\), you would shade everything outside the circle.

This practice makes it easy to identify solution sets at a glance.

• Ensure the inside isn’t shaded since the inequality is strict.
• Draw clear, distinct lines to separate shaded and non-shaded areas.

Shading offers a visual guide to understand and solve inequalities.

Origin-Centered Circles

An origin-centered circle has its center at \((0, 0)\). The equation \(x^2 + y^2 = r^2\) describes such a circle.

This type of circle is straightforward to work with because of its symmetry around the origin.

• All points at distance \(r\) from the origin lie on the circle.
• This symmetry makes equations and graphs involving these circles simpler to analyze.

Origin-centered circles are a central concept in coordinate geometry.

Radius of a Circle

The radius \(r\) of a circle is the distance from the center to any point on the circle. It is pivotal in determining the size of the circle.

For the circle equation \(x^2 + y^2 = r^2\), the radius \(r\) is the square root of the number on the right side. In this case:

• Since \(x^2 + y^2 = 36\), the radius is \(\sqrt{36} = 6\).
• Knowing \(r\) helps in accurately graphing the circle and addressing any related inequalities.

A clear understanding of radius enhances your ability to engage with geometric problems.