Problem 15 - Algebra and Trigonometry

Question

Systems of Equations and Inequalities. $$x^{2}+y^{2}>25$$

Step-by-Step Solution

Answer

The graph of the inequality \(x^{2}+y^{2}>25\) is a circle centered at the origin (0,0) of radius 5, and all the points outside this circle are shaded to indicate the solution set. The circle itself is not included in the solution set.

Step 1: Recognizing the formula

The equation \(x^{2}+y^{2}>25\) is a representation of a circle with radius 5. The inequality sign '>' means that we are looking at all points outside the circle with a radius of 5 units.

Step 2: Drawing the graph

The first step in graphing this inequality is to draw a circle with radius 5. It is centered at the origin (0,0). The inequality is 'greater than' (not 'greater than or equal to'), so the circle should be drawn with a dashed line instead of a solid one. This indicates that the points on the circle line are not included in the solution set.

Step 3: Filling the appropriate region

Since the inequality is \(x^{2}+y^{2}>25\), the solution set is all points outside the circle, not on the circle boundary or inside it. So, the region outside the circle is shaded.

Key Concepts

Understanding Circle Inequality

Circle inequalities involve expressions that describe regions in a coordinate plane relative to a circle. In the inequality \(x^2 + y^2 > 25\), this statement refers to a circle having a center at the origin \((0, 0)\) and a radius of \(5\) units. This is derived from the equation of a circle \(x^2 + y^2 = r^2\), where \(r\) is the radius. The inequality sign '> ' means we are interested in the set of points where \(x^2 + y^2\) are greater than \(25\). This describes the area outside of the circle, excluding the boundary itself which is implied by the lack of an '=' in the inequality.

Graphing Inequalities with Circles

Graphing inequalities that involve circles starts by drawing the circle itself based on the standard form \(x^2 + y^2 = r^2\). For \(x^2 + y^2 > 25\), we first draw a circle centered at \((0, 0)\) with a radius of \(5\). Unlike typical equations, when graphing inequalities, line markings change depending on the type of inequality. Since \(x^2 + y^2 > 25\) lacks an 'equal to' component, we use a *dashed* line for the circle. This indicates that the boundary is not part of the solution set. After establishing the boundary, shading is done outside the circle to represent all points that fulfill the inequality. Always remember: dashed line means 'without the boundary'. This graph visually represents where on the coordinate plane the inequality holds.

Solutions Sets of Inequalities

Inequality solution sets refer to collections of points that satisfy an inequality condition. For the inequality \(x^2 + y^2 > 25\), this means all points located outside the dashed circle we drew on the graph. These points are not merely the coordinates, but they signify a specific region. The visual representation helps to easily identify which coordinates make the inequality true, especially points that do not reside on or within the circle, but rather on the outside. When solving inequalities graphically, it's crucial to:

• Identify the boundary and whether it needs a dashed or solid line
• Choose a test point, like the origin, to determine which region to shade if it's not included in the circle
• Clearly shade the appropriate region to represent the solution set

Thus, inequality solution sets are not just about finding specific points, but highlighting regions that contain infinitely many solutions within their boundaries.