Problem 15 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$x^{2}+y^{2}>25$$

Step-by-Step Solution

Answer

The graph of the inequality \(x^{2}+y^{2}>25\) is a dashed circle with center at the origin and radius 5, with the area outside this circle shaded, indicating all the points that satisfy the inequality.

Step 1: Understand the inequality

This inequality \(x^{2}+y^{2}>25\) can be rewritten as \(x^{2}+y^{2}-25>0\), which is in the standard format of the equation for a circle: \(x^{2} + y^{2} = r^{2}\). This indicates a circle with center at the origin (0,0) and radius \(r = sqrt{25} = 5\). The inequality 'greater than' indicates that the solution will be all points outside this circle.

Step 2: Graph the circle

Draw a circle with center at the origin (0, 0) and radius 5 units. It's important to remember that the circle itself is not included in the solution because the inequality is 'greater than' and not 'greater than or equal to'. So, the circle should be drawn as a dashed line.

Step 3: Shade the solution area

Since the inequality is \(x^{2}+y^{2}>25\), the solution lies outside the circle. So, shade the area outside the dashed circle. Every point in this shaded area, if plugged into the inequality, would make the inequality true.

Key Concepts

Circle Equations

Understanding circle equations is essential when graphing inequalities involving circles. A circle's equation in the standard form is \(x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) is the center of the circle and \(r\) is the radius. In the given exercise, we have an inequality \(x^2 + y^2 > 25\), which implies a circle centered at the origin \( (0,0) \) with a radius of 5.

However, due to the strict inequality sign '>', we know that the solution doesn't include points on the circle, but rather points outside. The absence of the equal part in the inequality \(>\) means we represent the circle with a dashed line when sketching the graph. This dashed line serves as a boundary between the solution area and the rest of the coordinate plane. When graphing, remember to center your circle correctly and use the proper radius for accurate representation of the inequality.

Inequality Notation

Inequality notation plays an integral role in solving and graphing inequalities. It specifies the relationship between two values or expressions. Common inequality signs include \(>\) for 'greater than', \(<\) for 'less than', \(\geq\) for 'greater than or equal to', and \(\leq\) for 'less than or equal to'. When dealing with circle inequalities, remember that \(>\) or \(<\) dictate whether the solution area is outside or inside the circle, respectively.

Interpreting the Notation

In our problem \(x^2 + y^2 > 25\), the '>' sign indicates an open region; hence no solid line is drawn on the graph. If an equation contains \(\geq\) or \(\leq\), a solid line is drawn because the points on the line are included in the solution. Correct interpretation of inequality notation is crucial to accurately solving and graphing mathematical problems.

Shading Solution Areas

Shading solution areas is a visual method to represent all the solutions to an inequality on a graph. It's a powerful way to see at a glance which parts of the plane satisfy the inequality being considered. For the inequality \(x^2 + y^2 > 25\), the solution area consists of all points that are not on the circle and lie in the exterior region.

How to Shade Correctly

Ensuring that the shading correctly represents the solution set is paramount. Since we are dealing with a 'greater than' (\(>\)) situation in our exercise, we shade the area outside the dashed-line circle. Shading here tells us that any point in the shaded region is a valid solution to the inequality. It's vital to shade the correct side of the boundary to avoid confusion and accurately convey the solutions. Keeping shading consistent and clear is a critical factor for interpreting inequalities in graph form.