Solve The System Of Inequalities X^2-9y^2<9 And Y>2

Navigating the world of inequalities can be challenging, especially when dealing with systems of inequalities. This article provides a detailed walkthrough on how to solve the system of inequalities, focusing on the system:

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

We will explore the solution set and determine its properties, specifically addressing whether it is unbounded and whether a given point lies within the solution set. This guide aims to provide a clear and comprehensive understanding of the concepts and techniques involved in solving such systems.

Understanding the Inequalities

To effectively solve the system of inequalities, we must first understand each inequality individually. The first inequality, x² - 9y² < 9, represents a region bounded by a hyperbola. The second inequality, y > 2, represents the region above the horizontal line y = 2. Combining these two inequalities will give us the solution set for the system.

Analyzing the Hyperbolic Inequality

The inequality x² - 9y² < 9 can be rewritten in a standard form to better understand its graphical representation. Dividing both sides by 9, we get:

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

This is the equation of a hyperbola centered at the origin (0,0), with the transverse axis along the x-axis. The vertices of the hyperbola are at (±3, 0), and the asymptotes can be found by considering the equation x²/9 - y²/1 = 0, which gives us y = ±(1/3)x. The inequality x²/9 - y²/1 < 1 represents the region between the two branches of the hyperbola.

When graphing this inequality, it's essential to use a dashed line for the hyperbola itself because the inequality is strict (<), meaning points on the hyperbola are not included in the solution set. The region between the branches is the solution to this inequality, and we can verify this by testing a point, such as (0,0), which satisfies the inequality: 0² - 9(0²) < 9.

The hyperbola's graphical representation plays a crucial role in visualizing the solution set. Understanding the asymptotes and vertices helps in accurately sketching the hyperbola. The dashed lines indicate that the points on the hyperbola are not part of the solution, which is a critical detail when solving inequalities.

Examining the Linear Inequality

The second inequality, y > 2, is a linear inequality that represents all points above the horizontal line y = 2. This is a straightforward inequality to graph; simply draw a dashed horizontal line at y = 2 (since the inequality is strict) and shade the region above this line. The dashed line signifies that points on the line y = 2 are not part of the solution set.

The linear inequality provides a clear boundary for the solution set in the y-direction. It restricts the solution set to points where the y-coordinate is strictly greater than 2. This constraint is essential when finding the intersection with the hyperbolic inequality.

Graphing the System of Inequalities

To solve the system of inequalities, we need to graph both inequalities on the same coordinate plane and find the region where their solutions overlap. This overlapping region represents the solution set for the system.

Combining the Graphs

First, draw the hyperbola x²/9 - y²/1 = 1 with dashed lines and shade the region between its branches. Next, draw the horizontal line y = 2 with a dashed line and shade the region above it. The area where the shaded regions overlap is the solution set for the system of inequalities.

The graphical representation allows us to visualize the solution set. The intersection of the shaded regions shows all points (x, y) that satisfy both inequalities simultaneously. This visual approach is invaluable for understanding the nature of the solution set.

Identifying the Solution Set

The solution set is the region that is both between the branches of the hyperbola and above the line y = 2. This region is unbounded, meaning it extends infinitely in certain directions. Specifically, it extends upwards and outwards along the branches of the hyperbola.

The unbounded nature of the solution set is a significant characteristic. It indicates that there are infinitely many points that satisfy the system of inequalities, extending without limit in certain directions. This is a key observation when analyzing the solution set.

Analyzing the Solution Set

Now that we have graphed the system of inequalities and identified the solution set, we can analyze its properties. We will address two key questions:

1. Is the solution set unbounded?
2. Is the point (1,0) in the solution set?

Unbounded Solution Set

As observed from the graph, the solution set is indeed unbounded. The region extends infinitely upwards and outwards along the branches of the hyperbola. This means that there is no upper limit to the y-values in the solution set, and the region continues to expand outwards along the x-axis within the hyperbolic bounds.

The unbounded nature of the solution set can be analytically confirmed by noting that as x increases, y can also increase indefinitely while still satisfying both inequalities. This characteristic is inherent in the nature of the hyperbolic inequality combined with the linear inequality.

Checking the Point (1,0)

To determine whether the point (1,0) is in the solution set, we need to substitute x = 1 and y = 0 into both inequalities:

1. x² - 9y² < 9 becomes 1² - 9(0²) < 9, which simplifies to 1 < 9. This is true.
2. y > 2 becomes 0 > 2, which is false.

Since the point (1,0) does not satisfy the second inequality (y > 2), it is not in the solution set.

The verification process involves substituting the coordinates of the point into the inequalities. If the point satisfies all inequalities, it is part of the solution set. In this case, (1,0) fails to satisfy y > 2, thus it is excluded from the solution set.

Conclusion

In conclusion, to solve the system of inequalities:

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

we first analyzed each inequality individually, graphed them on the same coordinate plane, and identified the overlapping region as the solution set. We found that the solution set is unbounded and that the point (1,0) is not part of the solution set. This process demonstrates a comprehensive approach to solving systems of inequalities, combining algebraic understanding with graphical representation.

By thoroughly analyzing each inequality and their combined effect, we can accurately determine the solution set and its properties. This method is applicable to a wide range of systems of inequalities, making it a valuable tool in mathematical problem-solving.