Which Graph Represents The Solution Set For The Inequalities $x^2 + Y^2 < 25$ And $y^2 < 6x$?

Understanding inequalities and their graphical representations is a fundamental concept in mathematics, particularly when dealing with systems of inequalities. In this article, we will delve into the process of identifying the graphical solution of a system involving a circle and a parabola. Our specific focus will be on the system defined by the inequalities $x^2 + y^2 < 25$ and $y^2 < 6x$. This exploration will involve understanding the individual graphs of these inequalities and, more importantly, how to determine their common solution set. We'll break down the steps involved, from recognizing the equations to shading the correct regions on the coordinate plane. This detailed explanation aims to provide a clear understanding of how to tackle such problems, enabling you to confidently solve similar challenges in the future. So, let's embark on this mathematical journey and unravel the intricacies of graphing inequalities.

Deciphering the Inequalities: A Step-by-Step Approach

To effectively graph the solution of the system of inequalities $x^2 + y^2 < 25$ and $y^2 < 6x$, we need to break down each inequality individually and understand what region it represents on the coordinate plane. The key here is to recognize the standard forms of the equations associated with these inequalities and how they translate into graphical representations. Let's start with the first inequality, $x^2 + y^2 < 25$. This inequality bears a striking resemblance to the standard equation of a circle, which is $x^2 + y^2 = r^2$, where $r$ represents the radius of the circle. In our case, the equation $x^2 + y^2 = 25$ represents a circle centered at the origin (0, 0) with a radius of 5, since $r^2 = 25$ implies $r = 5$. However, the presence of the 'less than' sign (<) instead of an 'equals' sign (=) introduces a crucial distinction. The inequality $x^2 + y^2 < 25$ represents all the points inside the circle, not just the points on the circle itself. To depict this graphically, we draw a dashed circle to indicate that the boundary points are not included in the solution set, and then we shade the region inside the circle. Now, let's turn our attention to the second inequality, $y^2 < 6x$. This inequality represents a parabola. To better visualize it, we can consider the equation $y^2 = 6x$. This is a parabola that opens to the right, with its vertex at the origin (0, 0). The coefficient of the $x$ term determines the 'width' of the parabola; in this case, the parabola is wider than the standard parabola $y^2 = x$. Again, the 'less than' sign in the inequality $y^2 < 6x$ means that we are interested in the region inside the parabola, i.e., the region to the left of the curve. Similar to the circle, we draw a dashed parabola to indicate that the points on the parabola itself are not part of the solution, and we shade the region inside the parabola. By understanding these individual graphical representations, we are well-equipped to find the common solution set, which we will explore in the next section.

Finding the Overlapping Region: The Solution Set

Having deciphered the individual inequalities and their graphical representations, the next crucial step is to identify the overlapping region of the two inequalities, $x^2 + y^2 < 25$ and $y^2 < 6x$. This overlapping region represents the solution set to the system of inequalities, which means any point within this region satisfies both inequalities simultaneously. To visualize this, imagine overlaying the graph of the circle ($x^2 + y^2 < 25$) and the graph of the parabola ($y^2 < 6x$). The circle, as we established, is centered at the origin with a radius of 5, and the shaded region represents all points inside the circle. The parabola opens to the right, also with its vertex at the origin, and the shaded region represents all points to the left of the parabola. The overlapping region is the area where the shading from both graphs coincides. This region is bounded by both the circle and the parabola, creating a lens-like shape. It's important to note that the points where the circle and the parabola intersect are not included in the solution set because both boundaries are dashed, indicating strict inequalities (<). To accurately determine the solution set, you might want to sketch the graphs on a coordinate plane. Start by drawing the dashed circle and shading the interior. Then, draw the dashed parabola and shade the region to its left. The area where the shadings overlap is the visual representation of the solution set. In practical terms, this means that any point you pick within this overlapping region, when its coordinates are substituted into the original inequalities, will satisfy both inequalities. For instance, the point (1, 1) likely falls within this region. If we substitute these values into the inequalities, we get $1^2 + 1^2 = 2 < 25$ and $1^2 = 1 < 6(1) = 6$, which are both true. On the other hand, a point outside this region, such as (6, 0), would not satisfy both inequalities ($6^2 + 0^2 = 36$ is not less than 25). Therefore, the overlapping region is the definitive graphical representation of the solution to the system of inequalities.

Illustrative Examples: Solidifying the Concept

To further enhance your understanding, let's explore some illustrative examples that solidify the concept of graphing systems of inequalities. Consider a scenario where you need to determine if a specific point lies within the solution set of the inequalities $x^2 + y^2 < 25$ and $y^2 < 6x$. For instance, let's take the point (2, 3). To check if this point is a solution, we substitute $x = 2$ and $y = 3$ into both inequalities. For the first inequality, we have $2^2 + 3^2 = 4 + 9 = 13$, which is indeed less than 25. So, the point (2, 3) satisfies the first inequality. For the second inequality, we have $3^2 = 9$, and $6(2) = 12$. Since 9 is less than 12, the point (2, 3) also satisfies the second inequality. Therefore, the point (2, 3) lies within the solution set of the system. Now, let's consider a point outside the solution set, such as (4, 4). Substituting $x = 4$ and $y = 4$ into the inequalities, we get $4^2 + 4^2 = 16 + 16 = 32$, which is not less than 25. Even though $4^2 = 16$ is less than $6(4) = 24$, the point (4, 4) does not satisfy the first inequality, and therefore, it is not part of the solution set. Another illustrative example involves understanding the effect of changing the inequality signs. If the inequalities were $x^2 + y^2 > 25$ and $y^2 > 6x$, the solution set would be the region outside both the circle and the parabola. This means we would shade the areas that are not enclosed by either curve. Moreover, if the inequalities were $x^2 + y^2 ext{≤} 25$ and $y^2 ext{≤} 6x$, the boundaries (the circle and the parabola themselves) would be included in the solution set. We would represent this by drawing solid lines instead of dashed lines. These examples highlight the importance of careful consideration of the inequality signs and their impact on the graphical representation of the solution set. By working through such examples, you can develop a deeper intuition for graphing systems of inequalities.

Common Mistakes and How to Avoid Them

When graphing systems of inequalities like $x^2 + y^2 < 25$ and $y^2 < 6x$, there are several common mistakes that students often make. Recognizing these pitfalls and understanding how to avoid them can significantly improve your accuracy and problem-solving skills. One of the most frequent errors is incorrectly interpreting the inequality signs. As we've discussed, the 'less than' sign (<) indicates a region inside the curve (whether it's a circle or a parabola), while the 'greater than' sign (>) indicates a region outside the curve. A common mistake is to shade the wrong region, for example, shading the outside of the circle when the inequality is $x^2 + y^2 < 25$. To avoid this, always double-check the inequality sign and visualize which region it represents relative to the curve. Another common mistake is confusing strict inequalities (< or >) with non-strict inequalities (≤ or ≥). Strict inequalities mean that the boundary line (the circle or parabola itself) is not included in the solution set, so we use a dashed line to represent it. Non-strict inequalities mean the boundary line is included, so we use a solid line. Forgetting to use the correct type of line can lead to misinterpretation of the solution set. Another area where mistakes often occur is in accurately graphing the individual curves. For the circle $x^2 + y^2 < 25$, it's crucial to correctly identify the center and the radius. Similarly, for the parabola $y^2 < 6x$, knowing the direction of opening and the vertex is essential. A poorly drawn curve can significantly affect the accuracy of the solution set. To avoid these errors, take your time to plot the curves accurately, and double-check your calculations. Finally, one of the most critical steps in solving systems of inequalities is correctly identifying the overlapping region. This requires careful shading and visualization. A common mistake is to only shade one inequality or to misinterpret the area of intersection. To avoid this, it can be helpful to use different colors or shading patterns for each inequality and then clearly identify the region where they overlap. By being aware of these common mistakes and actively working to avoid them, you can confidently and accurately graph systems of inequalities.

Conclusion: Mastering the Art of Graphing Inequalities

In conclusion, mastering the art of graphing inequalities, particularly systems involving circles and parabolas like $x^2 + y^2 < 25$ and $y^2 < 6x$, is a crucial skill in mathematics. Throughout this article, we have meticulously dissected the process, starting from deciphering individual inequalities to identifying the overlapping solution set. We've emphasized the importance of recognizing the standard forms of equations, understanding the significance of inequality signs, and accurately representing these graphically. By recognizing that $x^2 + y^2 < 25$ represents the interior of a circle centered at the origin with a radius of 5 and that $y^2 < 6x$ represents the region inside a parabola opening to the right, we can begin to visualize the solution. The overlapping region, where the shaded areas of both inequalities coincide, visually represents the solution set – all points that satisfy both inequalities simultaneously. We've also highlighted common mistakes, such as misinterpreting inequality signs or incorrectly shading regions, and provided strategies to avoid them. By working through illustrative examples, we've reinforced the concept of how to determine if a specific point lies within the solution set. This involves substituting the point's coordinates into the inequalities and verifying whether both are satisfied. Ultimately, graphing systems of inequalities requires a blend of algebraic understanding and graphical visualization. It's not just about memorizing steps but about developing a conceptual understanding of what inequalities represent on the coordinate plane. By practicing and applying the techniques discussed in this article, you can confidently tackle graphing problems and further enhance your mathematical proficiency. So, embrace the challenge, hone your skills, and unlock the power of graphical solutions.