Solving Systems Of Inequalities X^2-9y^2<9 And Y>2

#1. Introduction

In the realm of mathematics, solving systems of inequalities is a fundamental skill, especially crucial in fields like optimization, economics, and engineering. This article aims to provide a comprehensive guide on tackling such systems, focusing on the specific example:

$\begin{array}{c} \left(x^2-9 y^2<9\right. \\ y>2'\n\end{array}$

We will not only dissect the solution process but also delve into the characteristics of the solution set, including whether it's bounded or unbounded and how to determine if a specific point belongs to the solution set. By the end of this guide, you'll have a solid understanding of how to approach similar problems and interpret the results effectively.

#2. Understanding the Inequalities

Before diving into the solution, let's first understand each inequality individually. This groundwork is crucial for visualizing the solution set and choosing the right approach. The first inequality, x^2 - 9y^2 < 9, represents a region bounded by a hyperbola. To see this more clearly, we can rewrite the inequality as:

$\frac{x^2}{9} - y^2 < 1$

This form reveals a hyperbola centered at the origin (0,0), with the transverse axis along the x-axis. The values 9 and 1 under the x^2 and y^2 terms, respectively, determine the shape and size of the hyperbola. Specifically, the hyperbola opens left and right, and the branches are defined by the asymptotes derived from the equation. The inequality < 1 indicates that the solution set includes all points inside the branches of the hyperbola, but not the hyperbola itself.

The second inequality, y > 2, is much simpler. It represents all points in the coordinate plane where the y-coordinate is greater than 2. This is a horizontal half-plane lying above the horizontal line y = 2. The line y = 2 itself is not included in the solution set because the inequality is strict (i.e., >). This means any point on the line y = 2 does not satisfy the inequality, but any point with a y-coordinate slightly greater than 2 does.

#3. Graphing the Inequalities

To visualize the solution set, we need to graph both inequalities on the same coordinate plane. Graphing is a powerful tool for understanding systems of inequalities, as it allows us to see the region that satisfies all the inequalities simultaneously.

3.1. Graphing the Hyperbola

The hyperbola $\frac{x^2}{9} - y^2 = 1$ has vertices at $(\pm 3, 0)$. The asymptotes can be found by setting the equation equal to 0: $\frac{x^2}{9} - y^2 = 0$, which gives $y = \pm \frac{x}{3}$. These asymptotes help define the shape of the hyperbola as it extends away from the vertices. When graphing the inequality $x^2 - 9y^2 < 9$, we draw the hyperbola as a dashed line to indicate that the points on the hyperbola are not included in the solution set. We then shade the region between the branches of the hyperbola, as this is where the inequality holds true. To confirm, we can test a point like (0,0), which lies between the branches. Substituting into the inequality, we get $0^2 - 9(0^2) < 9$, which simplifies to $0 < 9$, a true statement, confirming our shaded region is correct.

3.2. Graphing the Line

The inequality $y > 2$ represents all points above the horizontal line $y = 2$. When graphing this, we draw a dashed line at $y = 2$ to indicate that points on the line are not included in the solution. We then shade the region above the line, representing all points where the y-coordinate is greater than 2. Again, we can test a point, such as (0,3), which is above the line. Substituting into the inequality, we get $3 > 2$, a true statement, confirming our shaded region.

3.3. Identifying the Solution Set

The solution set of the system of inequalities is the region where the shaded areas from both inequalities overlap. This overlapping region represents all points that simultaneously satisfy both $x^2 - 9y^2 < 9$ and $y > 2$. It is the intersection of the region inside the hyperbola and the region above the line $y = 2$. This region will be unbounded, as it extends infinitely upwards within the confines of the hyperbola's branches.

#4. Analyzing the Solution Set

Now that we have a visual representation of the solution set, we can analyze its properties. This involves determining whether the solution set is bounded or unbounded and checking if specific points belong to the solution set. Analyzing the solution set provides deeper insights into the behavior of the system of inequalities.

4.1. Boundedness

Boundedness refers to whether the solution set can be enclosed within a finite region. In other words, is there a circle (or some other closed shape) that can contain the entire solution set? If such a circle exists, the solution set is bounded; otherwise, it is unbounded. In our case, the solution set is the intersection of the region inside the hyperbola $x^2 - 9y^2 < 9$ and the region above the line $y > 2$. While the hyperbola itself has boundaries, the region above the line $y > 2$ extends infinitely upwards. Therefore, the intersection of these regions also extends infinitely upwards, making the solution set unbounded. No matter how large a circle we draw, there will always be points within the solution set that lie outside the circle.

4.2. Point Inclusion

To determine if a specific point is in the solution set, we simply substitute the coordinates of the point into each inequality and check if both inequalities are satisfied. This is a straightforward way to verify whether a given point is a solution to the system.

For example, let's consider the point $(1, 0)$. Substituting into the first inequality, we get:

$1^2 - 9(0^2) < 9 \Rightarrow 1 < 9$

This inequality is true. However, substituting into the second inequality, we get:

$0 > 2$

This inequality is false. Since the point $(1, 0)$ does not satisfy both inequalities, it is not in the solution set. This aligns with our graphical understanding, as the point $(1, 0)$ lies below the line $y = 2$ and thus does not belong to the region defined by $y > 2$.

Now, let's consider a point that we expect to be in the solution set, such as $(4, 3)$. Substituting into the first inequality:

$4^2 - 9(3^2) < 9 \Rightarrow 16 - 81 < 9 \Rightarrow -65 < 9$

This inequality is true. Substituting into the second inequality:

$3 > 2$

This inequality is also true. Since $(4, 3)$ satisfies both inequalities, it is indeed in the solution set.

4.3. Additional Points to Consider

To further illustrate the concept of point inclusion, let's examine a few more points:

• (0, 3): Substituting into the inequalities, we get $0^2 - 9(3^2) < 9 \Rightarrow -81 < 9$ (true) and $3 > 2$ (true). Thus, (0, 3) is in the solution set.
• (5, 2.5): Substituting into the inequalities, we get $5^2 - 9(2.5^2) < 9 \Rightarrow 25 - 56.25 < 9 \Rightarrow -31.25 < 9$ (true) and $2.5 > 2$ (true). Thus, (5, 2.5) is in the solution set.
• (3, 2): Substituting into the inequalities, we get $3^2 - 9(2^2) < 9 \Rightarrow 9 - 36 < 9 \Rightarrow -27 < 9$ (true) and $2 > 2$ (false). Since the second inequality is false, (3, 2) is not in the solution set. This highlights the importance of strict inequalities ( > or < ) versus non-strict inequalities ( >= or <= ).

#5. Conclusion

Solving systems of inequalities involves understanding the individual inequalities, graphing them to visualize the solution set, and analyzing the properties of the solution set. In the given system, x^2 - 9y^2 < 9 and y > 2, the solution set is unbounded, meaning it extends infinitely in at least one direction. Furthermore, we've demonstrated how to check if a given point is in the solution set by substituting its coordinates into the inequalities and verifying that all inequalities are satisfied.

By mastering these techniques, you'll be well-equipped to tackle a wide range of problems involving systems of inequalities, a crucial skill in various mathematical and applied contexts. Understanding these concepts not only helps in solving problems but also provides a foundation for more advanced topics in mathematics and related fields.

#6. Key Takeaways

• Understanding Individual Inequalities: Before solving a system, make sure you fully grasp what each inequality represents graphically. This includes recognizing shapes like hyperbolas, lines, and circles, as well as understanding the difference between strict and non-strict inequalities.
• Graphical Representation: Graphing is a powerful tool for visualizing the solution set of a system of inequalities. The overlapping region of the individual inequality solutions represents the solution set of the system.
• Boundedness: Determine whether the solution set is bounded (can be enclosed in a finite region) or unbounded (extends infinitely). This can often be inferred from the graph.
• Point Inclusion: To check if a point is in the solution set, substitute its coordinates into each inequality. If all inequalities are satisfied, the point is in the solution set.
• Testing Points: When graphing, testing points can help confirm the shaded regions. Choose points that clearly lie within or outside the region you suspect is the solution.

By keeping these key takeaways in mind, you can approach systems of inequalities with confidence and accuracy. Practice is essential for mastering these skills, so be sure to work through a variety of examples to solidify your understanding.