Problem 45 - College Algebra

Question

For the following exercises, graph the system of inequalities. Label all points of intersection. $$x^{2}+3 y^{2}>16$$ $$3 x^{2}-y^{2}<1$$

Step-by-Step Solution

Answer

Graph two ellipses; check the regions where both inequalities are satisfied.

Step 1: Identify Each Inequality

First, we examine the given inequalities. The first inequality is \(x^2 + 3y^2 > 16\) and the second is \(3x^2 - y^2 < 1\). These can be seen as representing non-standard ellipses.

Step 2: Standard Form Conversion

Convert each inequality to a format that can be visually represented. For the first inequality, consider \(x^2/16 + y^2/5.33 > 1\), which describes the region outside an ellipse centered at the origin. For the second, consider \(x^2/0.33 - y^2 < 1\), which describes another rotated region that looks elliptical.

Step 3: Graph the Region for Each Inequality

Draw each boundary. Graph \(x^2 + 3y^2 = 16\) as an ellipse centered at the origin stretched in the y-direction. Graph \(3x^2 - y^2 = 1\) similar to an ellipse, noting areas inside are solutions. Use a dotted line for boundaries, since inequalities are strict.

Step 4: Determine Intersection Points

Calculate potential points of intersection by solving \(x^2 + 3y^2 = 16\) and \(3x^2 - y^2 = 1\) simultaneously. Substitute to find common solutions. Solve these equations using substitution or elimination methods to find exact (x, y) points.

Step 5: Label Intersection Points

Label any valid intersection points found from solving the simultaneous equations directly on the graph. Remember that for non-equal cases, these points are not part of the solution.

Key Concepts

System of Inequalities

A **system of inequalities** involves multiple inequalities that are considered simultaneously. Each inequality represents a region in the coordinate plane, and the solution to the system is where these inequalities overlap.
This means we are looking for the area in the graph where all inequalities are true at the same time.

Consider two inequalities, such as:

• \( x^2 + 3y^2 > 16 \)
• \( 3x^2 - y^2 < 1 \)

These inequalities graph regions that might overlap in complex ways. To solve such a system, each inequality is first visualized individually. Then we identify the common region where both inequalities hold.
It's like finding where two different colored shapes on a transparent sheet overlap when placed on top of one another.
Be mindful that the solution to the system is often not just a point, but rather a whole region encompassing several points.

Intersection Points

**Intersection points** occur where the graphs of two equations or inequalities meet. These are typically points in the coordinate plane where two graphs cross each other.
These points are critical because they help us determine the boundary of the overlapping solution region of a system of inequalities.

To find intersection points, solve the equations formed by making the inequalities equal:

• \( x^2 + 3y^2 = 16 \)
• \( 3x^2 - y^2 = 1 \)

By using methods such as substitution or elimination, we identify the values of \( x \) and \( y \) that satisfy both equations simultaneously.
These points are marked directly on the graph, as they serve as potential boundaries of intersection in the visualization of solutions.

Ellipses

An **ellipse** is a shape that looks like a flattened circle and can be described by equations with quadratic terms. In the standard form, its equation appears as:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Where \( a \) and \( b \) determine the lengths of the axes, and the ellipse is centered at the origin.

For inequalities like \( x^2 + 3y^2 > 16 \), we visualize regions outside the ellipse:\[ (x^2/16) + (y^2/5.33) > 1 \] gives us a region outside the boundary of a standard ellipse.
For \( 3x^2 - y^2 < 1 \), although it isn't a pure ellipse, its shape resembles one, forming another distinct region on the graph.
These ellipses help in understanding where solutions lie relative to these boundaries. Sometimes ellipses overlap or share common areas with similar or different geometrical figures.

Coordinate Geometry

**Coordinate geometry**, also known as analytic geometry, is the study of geometrical figures using a coordinate system. It involves representing equations and inequalities in a two-dimensional plane with x and y axes.

In coordinate geometry, each point in the plane is defined by an ordered pair \((x, y)\).
For the given system of inequalities:

• The inequality \( x^2 + 3y^2 > 16 \) describes points outside an ellipse.
• The inequality \( 3x^2 - y^2 < 1 \) describes points inside another region.

Graphing these involves plotting the curves of each equality (where the inequalities are replaced with equal signs) and shading the appropriate regions.
Through this graphical strategy, you can effectively visualize and solve complex problems involving multiple shapes and their intersections, enhancing understanding of spatial relations in math.