Problem 9 - Algebra A Combined Function

Question

Graph each inequality. $$ x^{2}+y^{2} \leq 9 $$

Step-by-Step Solution

Answer

The inequality represents a solid circle of radius 3, centered at the origin, with shading inside the circle.

Step 1: Identify the Shape

The inequality \( x^2 + y^2 \leq 9 \) represents a circle with a radius. We need to rewrite it in a form \( x^2 + y^2 \leq r^2 \) where \( r \) is the radius.

Step 2: Determine the Radius

The equation \( x^2 + y^2 \leq 9 \) suggests the circle is centered at the origin \((0,0)\) with \( r^2 = 9 \), so the radius \( r = 3 \).

Step 3: Draw the Circle

Draw a circle centered at the origin with a radius of 3. Since the inequality is \( \leq \), the circle is solid, indicating points on the circle are included.

Step 4: Shade the Inside Area

Since the inequality is \( x^2 + y^2 \leq 9 \), this means any point inside or on the circle satisfies the inequality. Shade the area inside the circle to indicate all these solutions.

Key Concepts

Inequalities in Two Variables

Inequalities involving two variables, like \(x^2 + y^2 \leq 9\), represent a relationship where the values of \(x\) and \(y\) do not satisfy a single solution. Instead, they encompass a range of values. When dealing with such inequalities, we treat them graphically to identify all the possible solutions visually.

• The circle equation \(x^2 + y^2 = r^2\) becomes an inequality \(x^2 + y^2 \leq r^2\), indicating that the solutions are not just the edge of the circle (perimeter) but include everything inside or on the circle.
• The inequality \(\leq\) or \(<\) specifies whether the edge is included (solid line) or excluded (dotted line).

By looking at such inequalities through the lens of geometry, we simplify the visualization and understanding of potential solutions for \((x, y)\).

Graphing Circles

Graphing a circle starts with recognizing it in the standard form equation \(x^2 + y^2 = r^2\). For the inequality \(x^2 + y^2 \leq 9\), it signifies a circle with center at the origin and a radius determined by the square root of 9, which is 3.

• The inequality \(\leq\) ensures the circle is solid, meaning all points on and inside the circle are included.
• Graphically, draw the circle using compass tools or graphing software by maintaining a fixed radius from the center point \((0, 0)\).

Encompassing the space inside the boundary caters to all permissible solutions, making sure to visually delineate the entire inequality area effectively.

Coordinate Geometry

Coordinate geometry serves as the mathematical bridge between algebra and geometry, translating equations into visual representations. For circles expressed as \(x^2 + y^2 \leq r^2\), coordinate geometry helps in:

• Determining the center and radius from the equation to accurately plot the shape.
• Using axes to measure distances, ensuring precision in drawing the circle.
• Deciphering transformations, such as translating or expanding shapes, that affect the inequality.

When working with this inequality, plot the circle's center at origin, always checking against measurements to ensure an accurate graph depiction.

Solutions of Inequalities

The solutions to an inequality like \(x^2 + y^2 \leq 9\) are the sets of all point pairs \((x, y)\) within the defined region. To confirm whether a point fits the inequality:

• Substitute the point’s \(x\) and \(y\) values into the inequality to check satisfaction.
• Points on the circle's perimeter satisfy the equality \(x^2 + y^2 = 9\), meaning they are part of the solution set.
• Any point within, where \(x^2 + y^2 < 9\), also satisfies the inequality.

This inclusive approach underscores why the region inside a plotted circle is shaded, visibly representing all solutions to observers.