Problem 45 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

Graph the ellipse and hyperbola, shading the appropriate regions for the inequalities. Feasible region points of intersection must be visually or algebraically determined.

Step 1: Understand the Inequalities

The given system consists of two inequalities. The first is \( x^2 + 3y^2 > 16 \) which represents an ellipse with a vertical major axis. The second is \( 3x^2 - y^2 < 1 \) which represents a hyperbola that opens along the x-axis.

Step 2: Rewrite the Inequalities for Graphing

Rewrite the inequalities in standard form for graphing. The first inequality can be expressed as \( \frac{x^2}{16} + \frac{y^2}{rac{16}{3}} = 1 \), and the second as \( \frac{x^2}{rac{1}{3}} - \frac{y^2}{1} = 1 \). These forms help in identifying the vertices and asymptotes.

Step 3: Graph the Ellipse

Draw the ellipse \( \frac{x^2}{16} + \frac{y^2}{rac{16}{3}} = 1 \). It has a center at (0,0), semi-major axis length of \( 4 \) along the y-axis, and semi-minor axis length of \( 4\sqrt{3}/3 \) along the x-axis. Shade the region outside the ellipse since the inequality is \( x^2 + 3y^2 > 16 \).

Step 4: Graph the Hyperbola

Draw the hyperbola \( \frac{x^2}{rac{1}{3}} - \frac{y^2}{1} = 1 \). The center is at (0,0), vertices are at (\( \sqrt{3},0 \)) and (\( -\sqrt{3},0 \)), asymptotes are lines passing through the origin with slopes \( \pm \sqrt{3} \). Shade the region inside the hyperbola as the inequality is \( 3x^2 - y^2 < 1 \).

Step 5: Identify the Feasible Region

The feasible region is where the shaded regions from the ellipse's exterior and the hyperbola's interior overlap. Determine the approximate points of intersection visually or through solving the related equality structures.

Step 6: Label Points of Intersection

The intersection points will be where the shapes intersect under their respective inequality restrictions. Calculate these precisely if possible, though numerical approximations may also be used.

Key Concepts

Understanding the Ellipse

An ellipse is a smooth, closed curve that resembles an elongated circle. Mathematically, it can be expressed in the form of an inequality or equation that involves both x and y variables. The standard form of an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, "a" and "b" are the lengths of the semi-major and semi-minor axes, respectively.

• The major axis is the longest diameter that runs through the center.
• The minor axis is the shortest diameter perpendicular to the major axis through the center.

For the inequality \( x^2 + 3y^2 > 16 \), we rearranged and identified it as an ellipse with a vertical major axis. This is because the term with "y" has a larger multiplier \((3)\) compared to "x". Thus, in graphing this ellipse, we focus on plotting and shading the region outside the elliptical boundary.

Understanding the Hyperbola

A hyperbola, like an ellipse, involves both x and y variables but has a fundamental difference in shape and definition. The standard equation of a hyperbola centered at the origin looks like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \). It has two disconnected curves, or branches, which mirror each other.

• The vertices are key points on the curves connected by the transverse axis.
• Asymptotes are straight lines that approach the curves but do not intersect them.

For our inequality system, \( 3x^2 - y^2 < 1 \), the hyperbola opens along the x-axis because of the negative sign attributed to the y-term. The vertices at \((\pm \sqrt{3}, 0)\) and asymptotes with slopes \( \pm \sqrt{3} \) are crucial for graphing an accurate representation. Unlike the ellipse, we shade the region inside the hyperbolic boundary.

Finding Intersection Points

Intersection points are where two curves meet or cross over in the coordinate plane. For our system of inequalities, these are critical to determine for accurately graphing and understanding the solution set. To find the intersection points analytically, we would solve the equal portions of each inequality together, effectively treating them as equations.

• Use substitution or elimination to solve the equations for points of intersection.
• Check solutions against both inequalities to ensure they satisfy the system.

In this case, visual approximation can also be helpful, as exact analytical solutions could become complex or lengthy. Intersection points define the limits and characteristics of the feasible region in graphing.

Working with Inequalities

In the context of graphing, inequalities define the range or region a solution can occupy in the coordinate plane. When dealing with inequalities:- You graph not just a line or curve, but a whole area.- The sign (> or <) tells whether to shade above or below (or inside or outside) the boundary.For instance:- With \( x^2 + 3y^2 > 16 \), you shade outside the ellipse because solutions lie beyond the curve.- For \( 3x^2 - y^2 < 1 \), you shade inside the hyperbola, capturing where solutions validly exist.

• The boundary line is often a dotted line to indicate that the points on it are not solutions.
• A solid line is used if the inequality includes equality (like \( \leq \) or \( \geq \)).

Recognizing these conventions ensures clear and communicative graph representations.