Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, graph the inequality. $$ x^{2}+y^{2}<4 $$

Step-by-Step Solution

Answer

The graph is a dashed circle of radius 2 centered at the origin, shading the inside region.

Step 1: Identify the Type of Graph

The given inequality is similar to the equation for a circle: \(x^2 + y^2 = 4\). This represents a circle centered at the origin (0,0) with a radius of 2. The inequality \(x^2 + y^2 < 4\) indicates the area inside the circle.

Step 2: Determine the Boundary of the Circle

Rewrite the boundary equation \(x^2 + y^2 = 4\) to understand it's a circle with radius 2. Thus, plot the circle with its center at (0,0) and mark the points where the circle intersects the x-axis at (-2,0) and (2,0) and the y-axis at (0,-2) and (0,2).

Step 3: Decide Line Type for Boundary

Since the inequality is strict (\(<\)), we use a dashed line to represent the boundary of the circle. This indicates that points on the circle are not included in the solution set.

Step 4: Shade the Appropriate Region

Since the inequality is \(<\), we shade the entire region inside the dashed circle. This shading represents all points \((x,y)\) such that \(x^2 + y^2 < 4\), which means all points inside the circle but not on the boundary.

Step 5: Verify with Test Point

To confirm the correct region is shaded, choose a test point from inside the circle, such as (0,0), and substitute into the inequality: \(0^2 + 0^2 < 4\). This simplifies to \(0 < 4\), which is true, confirming the area within the circle is correctly shaded.

Key Concepts

Understanding Circles in Graphing

In the world of graphing, circles are a special type of geometric shape that are defined by a specific equation. A standard equation for a circle is given by \[ (x - h)^2 + (y - k)^2 = r^2 \] where \( (h, k) \) represents the center of the circle, and \( r \) is the radius. The equation signifies that every point on the circle is exactly \( r \) units away from the center.
This concept is crucial when graphing inequalities involving circles. When you encounter an inequality like \( x^2 + y^2 < 4 \), it's similar to the circle equation \( x^2 + y^2 = 4 \). Here, the circle is centered at the origin (0,0) with a radius of 2 since \( 4 = 2^2 \). Recognizing this pattern helps understand that the inequality describes all points inside this circle.

Deciphering Inequality Solution Sets

Inequality solution sets can seem tricky, but with a little practice, they become easier to manage. When you have an inequality involving a circle, such as \( x^2 + y^2 < 4 \), it defines a region.

• The solution set includes all points (x, y) that make the inequality true.
• For \( x^2 + y^2 < 4 \), it includes all points inside the circle defined by \( x^2 + y^2 = 4 \) but excludes points on the circle itself due to the "<" symbol.
• If it were \( \leq \), the points on the circle would be included, hence labeled a solution set with a solid boundary.

To represent a strict inequality like this one, use a dashed line for the circle's boundary, showing that it's not part of the solution. Then, shade the interior to portray all the solution points.

Graphing on the Coordinate Plane

When graphing inequalities on the coordinate plane, it's important to follow clear steps. The coordinate plane acts as a grid where every point corresponds to a pair of numbers \((x, y)\). When you plot an inequality such as \( x^2 + y^2 < 4 \), you'll:
1. Begin by sketching the boundary. For \( x^2 + y^2 = 4 \), draw a circle centered at the origin with a radius of 2. Mark intersections with the axes at points like (-2,0), (2,0), (0,-2), and (0,2).
2. Use a dashed line to depict the circle's boundary. This dashed line symbolizes that the circle itself isn't part of the solution.
3. Identify the area to shade. Since the inequality is "less than," shade the entire region within the circle.
4. Optionally, confirm your shading by picking a test point inside the circle, like (0,0). If substituting it into the inequality makes a true statement, your shading is likely correct.
These steps help create a clear visual representation of inequality graph solutions, ensuring accuracy and understanding.