Graphing Solutions For Inequalities Y ≤ X² + 1 And X > Y² – 5

In order to graph the solution of the inequalities y ≤ x² + 1 and x > y² – 5, we first need to understand what each inequality represents graphically. The first inequality, y ≤ x² + 1, involves a parabola. Specifically, it represents all the points on or below the parabola defined by the equation y = x² + 1. This parabola opens upwards, with its vertex at the point (0, 1). Key points like (-2, 5) and (2, 5) also lie on this parabola, helping to define its shape more accurately. This parabola is a fundamental concept in algebra and calculus, representing a quadratic function whose graph has a distinctive U-shape. Understanding the properties of parabolas, such as their vertex, axis of symmetry, and direction of opening, is crucial for graphing them correctly. In this case, the parabola opens upwards, meaning that its arms extend infinitely upwards from the vertex. The inequality y ≤ x² + 1 includes all points below or on the parabola, indicating that the solution region lies within and on the curve. Recognizing the role of the inequality sign is essential, as it determines whether the boundary (the parabola itself) is included in the solution set. For instance, if the inequality were y < x² + 1, the parabola would be drawn with a dashed line to indicate that points on the curve are not part of the solution. However, in our case, the solid line implies that all points on the parabola are included in the solution set. By understanding these properties, we can accurately represent the inequality y ≤ x² + 1 graphically and proceed to analyze the second inequality.

The second inequality, x > y² – 5, also represents a parabola, but this time it opens to the right. The equation x = y² – 5 defines a parabola that opens along the x-axis instead of the y-axis. Its vertex is at the point (-5, 0), and it passes through points such as (-4, -1) and (-4, 1). The inequality x > y² – 5 includes all points to the right of this parabola. This orientation of the parabola is crucial to understand because it differs from the typical vertical parabolas often encountered in introductory algebra. The fact that the x-term is isolated and the y-term is squared indicates that the parabola will open horizontally. The vertex at (-5, 0) provides a starting point for sketching the graph, and knowing that it opens to the right helps in visualizing the region that satisfies the inequality. Furthermore, the points through which the parabola passes, such as (-4, -1) and (-4, 1), provide additional guidance for accurately drawing the curve. These points help to ensure that the shape and position of the parabola are correctly represented. The inequality x > y² – 5 means that only the points to the right of the parabola, but not on it, are included in the solution set. This is because the inequality does not include an equals sign. If it were x ≥ y² – 5, the parabola itself would also be part of the solution. Understanding the distinction between strict and non-strict inequalities is crucial for correctly interpreting the graph. By carefully considering the properties of this horizontal parabola, including its vertex, direction of opening, and the points it passes through, we can accurately represent the inequality x > y² – 5 on the coordinate plane.

To graph the solution for y ≤ x² + 1, first, draw the parabola y = x² + 1. This parabola opens upwards with its vertex at (0, 1). Since the inequality includes “less than or equal to,” the parabola is drawn with a solid line to indicate that the points on the parabola are part of the solution. The points (-2, 5) and (2, 5) are also on this parabola and help in sketching it accurately. Shading the region below the parabola represents all points where y is less than or equal to x² + 1. This shading is a crucial step in visually representing the solution set. By shading below the parabola, we are highlighting all the points on the coordinate plane that satisfy the inequality. The solid line of the parabola itself is also included in the solution, as the inequality includes the “equal to” condition. This means that any point on the parabola also makes the inequality true. The accuracy of the shaded region is essential for correctly identifying the final solution, which will be the intersection of the solutions of both inequalities. Therefore, it is important to carefully shade the region below the parabola, ensuring that it extends indefinitely downwards, as the inequality holds for all such points. By understanding and correctly graphing the first inequality, we set the stage for combining it with the second inequality to find the overall solution.

Next, to graph the solution for x > y² – 5, draw the parabola x = y² – 5. This parabola opens to the right, with its vertex at (-5, 0). Since the inequality is “greater than,” the parabola is drawn with a dashed line to indicate that the points on the parabola are not part of the solution. The points (-4, -1) and (-4, 1) are on this parabola and help in sketching it accurately. Shade the region to the right of the parabola, representing all points where x is greater than y² – 5. The dashed line for this parabola is a critical distinction from the solid line used for the previous parabola. It signifies that the points on the parabola x = y² – 5 do not satisfy the strict inequality x > y² – 5. The shading to the right of the parabola visually represents all the points that make the inequality true. This region extends indefinitely to the right, encompassing all points where the x-coordinate is greater than the value of y² – 5. The accuracy of this shading is crucial for the next step, which involves finding the overlapping region that satisfies both inequalities. The dashed line and the shading work together to provide a clear visual representation of the solution set for this inequality. By carefully drawing the parabola and shading the appropriate region, we prepare to identify the common solution area when combined with the graph of the first inequality.

The solution region is where the shaded regions of both inequalities overlap. This region represents all the points that satisfy both y ≤ x² + 1 and x > y² – 5 simultaneously. The overlapping region is a critical visual element that represents the set of all points that satisfy both inequalities. This area is the intersection of the two shaded regions, highlighting the points where the conditions of both inequalities are met. Points within this region have y-values that are less than or equal to x² + 1 and x-values that are greater than y² – 5. To accurately identify this region, it is essential to have clearly shaded the areas for each inequality. The overlap will then be visually apparent, bounded by the two parabolas. Understanding that the solution region is where both inequalities hold true is fundamental to solving systems of inequalities graphically. This method provides a visual representation of the solution set, making it easier to comprehend the relationship between the inequalities. The overlapping region may be a bounded area or extend infinitely, depending on the nature of the inequalities. In this case, the region is bounded by the two parabolas, creating a distinct shape that represents the solution set. By focusing on the overlapping region, we can pinpoint the exact set of points that satisfy both given conditions.

In this case, the solution region is the area bounded by the upward-opening parabola y = x² + 1 and the rightward-opening parabola x = y² – 5. The boundary of the region includes the parabola y = x² + 1 (since the inequality is less than or equal to) but does not include the parabola x = y² – 5 (since the inequality is strictly greater than). This distinction is crucial for accurately defining the solution set. The solid line of the parabola y = x² + 1 indicates that the points on this curve are part of the solution, while the dashed line of the parabola x = y² – 5 indicates that the points on this curve are not. The shape of the solution region is defined by the intersection of the two parabolas, creating a specific geometric area on the coordinate plane. This area represents all the points that simultaneously satisfy both inequalities. To fully understand the solution, one might test points within and outside the region to confirm that they either satisfy both inequalities or violate at least one. The visual representation of the solution region provides a clear and intuitive understanding of the combined constraints imposed by the two inequalities. By correctly identifying the bounded area and noting which boundaries are included, we can accurately describe the set of points that form the solution.

Therefore, the graphical solution is the region where the inequalities y ≤ x² + 1 and x > y² – 5 overlap, clearly showing all points that satisfy both conditions. This graphical method provides a comprehensive and visual way to understand the solution set of a system of inequalities. By graphing each inequality separately and then identifying the overlapping region, we can determine all the points that meet the conditions of both inequalities. The process involves understanding the shapes of the curves represented by the inequalities, in this case, parabolas, and accurately shading the regions that correspond to the inequality conditions. The final solution is the intersection of these shaded regions, representing all the points that satisfy both inequalities simultaneously. This approach is particularly useful for complex inequalities where algebraic methods might be cumbersome. The graphical solution provides an intuitive and clear representation of the solution set, making it easier to understand and communicate. By mastering this technique, one can effectively solve a wide range of inequality problems and gain a deeper understanding of their solutions. In summary, the graphical solution method is a powerful tool for analyzing and solving systems of inequalities, providing both visual insight and a comprehensive understanding of the solution set.

Unfortunately, I can't draw the graph here, but imagine the upward-opening parabola y = x² + 1 drawn with a solid line and the region below it shaded. Then, imagine the rightward-opening parabola x = y² – 5 drawn with a dashed line and the region to its right shaded. The solution is the area where these shaded regions overlap.