Problem 14 - Intermediate Algebra

Question

Graph each inequality. See Examples 1 and \(2 .\) $$ \frac{x^{2}}{25}+\frac{y^{2}}{4} \geq 1 $$

Step-by-Step Solution

Answer

Graph the ellipse and shade everything outside of it, including the ellipse itself.

Step 1: Identify the Standard Form

The given inequality \( \frac{x^2}{25} + \frac{y^2}{4} \geq 1 \) represents an ellipse in its standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \( a^2 = 25 \) and \( b^2 = 4 \). Thus, \( a = 5 \) and \( b = 2 \).

Step 2: Determine the Boundary

The boundary of the solution is the ellipse itself given by the equation \( \frac{x^2}{25} + \frac{y^2}{4} = 1 \). This ellipse is centered at the origin (0,0), with the major axis along the x-axis of length \( 2a = 10 \) and the minor axis along the y-axis of length \( 2b = 4 \).

Step 3: Graph the Boundary

Draw the ellipse using the intercepts. The ellipse intercepts the x-axis at \( x = \pm 5 \) and the y-axis at \( y = \pm 2 \). Plot these points and sketch the ellipse centered at (0,0).

Step 4: Determine the Inequality Region

The inequality \( \frac{x^2}{25} + \frac{y^2}{4} \geq 1 \) means we are interested in the region outside or on the boundary of the ellipse. This includes all points (x,y) for which \( \frac{x^2}{25} + \frac{y^2}{4} > 1 \) as well as \( \frac{x^2}{25} + \frac{y^2}{4} = 1 \).

Step 5: Shading the Correct Region

Shade the region outside the ellipse to represent all x and y values satisfying the inequality. Include the boundary of the ellipse since the inequality is \( \geq \).

Key Concepts

Ellipse Inequality

An ellipse inequality describes a region in a coordinate plane that includes an ellipse and sometimes the space either inside or outside of it. The inequality \( \frac{x^2}{25} + \frac{y^2}{4} \geq 1 \) tells us that we are dealing with an ellipse because of its standard ellipse form similarities. For inequalities, they will tell us more than just where the ellipse is. This one, for instance, includes regions on or outside the boundary of the ellipse.

• The equation, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), is the standard form for an ellipse.
• A given inequality that involves an ellipse is more than just that boundary; it defines regions on the graph.
• The symbol used in the inequality (\( \geq \), \(< \), etc.) will instruct you which side of the boundary matters.

Understanding these concepts can help in accurately visualizing and graphing any ellipse inequalities.

Coordinate Plane

The coordinate plane is a two-dimensional surface that you use for graphing mathematical figures like ellipses. The plane has two perpendicular axes, usually labeled as the x-axis (horizontal) and the y-axis (vertical). Every point on this plane can be identified using ordered pairs \((x, y)\), where 'x' indicates how far to go horizontally, and 'y' describes the vertical movement from the origin \((0, 0)\).

• The origin is the point \((0, 0)\), where both axes meet.
• Positive values of 'x' are to the right, and negative to the left.
• Positive 'y' values go upwards, and negative go downwards.

Using the coordinate plane allows you to visually understand the location and shape of geometric figures, like the ellipse for our inequality.

Inequality Graphing

Graphing inequalities involves plotting a boundary line or curve and then shading the appropriate region that satisfies the inequality. For our exercise, we dealt with an inequality involving an ellipse.
Steps in Graphing Inequalities:

• First, **sketch the boundary**. Here, it's the ellipse from the equation \( \frac{x^2}{25} + \frac{y^2}{4} = 1 \).
• **Identify the region**: Since the inequality is \( \geq 1 \), this includes the area outside or on the ellipse.
• **Shade the correct area** that represents the solution set. In this exercise, that's the area outside the ellipse including its boundary.

The graph gives a visual cue of all possible solutions to the inequality, aiding in understanding and interpreting the problem better.

Conic Sections

A conic section is a curve obtained by intersecting a plane with a double cone. An ellipse is one example of a conic section. When discussing conic sections, four main types can emerge depending on how the plane slices the cone: circle, ellipse, parabola, and hyperbola.
Ellipses:

• Are closed curves, like flattened circles.
• Have two principal axes: major (longest) and minor (shortest).
• All points on an ellipse are at a constant sum of distances from two fixed points called foci.

Understanding these basic properties of conic sections allows you to see the connection between the algebraic representation (like our inequality) and the geometry of these curves.