Does The Coordinate Point (1, 8) Satisfy The Inequality Y ≤ 2x + 6?

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Introduction

In the realm of mathematics, inequalities play a crucial role in defining relationships between variables and regions on a coordinate plane. These mathematical statements, which compare two expressions using symbols like '≤' (less than or equal to), '≥' (greater than or equal to), '<' (less than), or '>' (greater than), are fundamental to various fields, including algebra, calculus, and optimization. Understanding how to determine whether a specific point satisfies an inequality is a key skill in these areas. In this article, we will delve into the process of verifying whether a given point satisfies a particular inequality. Specifically, we will focus on the inequality y ≤ 2x + 6 and investigate whether the point (1, 8) lies within the solution set defined by this inequality.

This exploration is not merely an abstract exercise; it has practical implications in diverse applications. For instance, in economics, inequalities can represent budget constraints, limiting the combinations of goods and services a consumer can afford. In engineering, they can define tolerance limits, ensuring that manufactured parts meet specific quality standards. In computer science, inequalities are used in optimization algorithms to find the best solution within a set of constraints. Therefore, mastering the ability to work with inequalities and determine solution sets is essential for anyone pursuing a career in these fields.

Our journey will begin with a step-by-step explanation of the method for verifying whether a point satisfies an inequality. We will then apply this method to the specific case of the point (1, 8) and the inequality y ≤ 2x + 6. By substituting the coordinates of the point into the inequality, we will be able to determine whether the inequality holds true. Finally, we will discuss the implications of our findings and explore the graphical representation of the inequality, providing a visual understanding of the solution set and the position of the point (1, 8) within it.

Understanding Inequalities

Before we dive into the specifics of the point (1, 8) and the inequality y ≤ 2x + 6, it's essential to have a solid grasp of what inequalities represent. An inequality, unlike an equation, does not define a single solution but rather a range of solutions. This range can be visualized as a region on a coordinate plane. For example, the inequality y ≤ 2x + 6 represents all the points (x, y) that lie on or below the line defined by the equation y = 2x + 6. This region extends infinitely in the downward direction, encompassing all points where the y-coordinate is less than or equal to 2 times the x-coordinate plus 6.

The line y = 2x + 6 itself is a boundary, separating the region where the inequality holds true from the region where it does not. Points that lie on the line are included in the solution set because the inequality includes the 'equal to' condition (≤). If the inequality were y < 2x + 6, the line would be a dashed line, indicating that points on the line are not part of the solution set.

The concept of a solution set is crucial when working with inequalities. The solution set is the collection of all points that satisfy the inequality. In the case of y ≤ 2x + 6, the solution set is an infinite region below the line y = 2x + 6. Determining whether a point satisfies an inequality is equivalent to checking if the point belongs to the solution set. This involves substituting the coordinates of the point into the inequality and verifying if the resulting statement is true.

Inequalities can also be more complex, involving multiple variables or non-linear expressions. However, the fundamental principle remains the same: an inequality defines a range of possible solutions, and checking if a point satisfies the inequality involves substituting the point's coordinates and verifying the resulting statement. This understanding forms the basis for our analysis of the point (1, 8) and the inequality y ≤ 2x + 6.

The Method for Verifying a Point in an Inequality

Verifying whether a point satisfies an inequality is a straightforward process that involves substituting the coordinates of the point into the inequality and evaluating the resulting statement. This method is applicable to inequalities in any number of variables, although we will focus on two-variable inequalities for this particular problem. The steps involved in this method are as follows:

  1. Identify the Inequality: The first step is to clearly identify the inequality you are working with. In our case, the inequality is y ≤ 2x + 6. This inequality establishes a relationship between the variables x and y, defining a region on the coordinate plane.
  2. Identify the Point: Next, identify the point you want to test. In this case, the point is (1, 8). This point has an x-coordinate of 1 and a y-coordinate of 8. We want to determine whether this specific point lies within the region defined by the inequality.
  3. Substitute the Coordinates: This is the core of the method. Substitute the x-coordinate of the point for x in the inequality and the y-coordinate of the point for y in the inequality. In our case, we substitute x = 1 and y = 8 into the inequality y ≤ 2x + 6. This gives us 8 ≤ 2(1) + 6.
  4. Simplify and Evaluate: After substituting the coordinates, simplify the expression on both sides of the inequality. In our case, we have 8 ≤ 2(1) + 6. Simplifying the right side gives us 8 ≤ 2 + 6, which further simplifies to 8 ≤ 8.
  5. Determine if the Inequality Holds True: The final step is to determine whether the resulting statement is true. In our case, we have 8 ≤ 8. Since 8 is indeed less than or equal to 8, the inequality holds true. If the resulting statement were false (e.g., 8 < 8), then the inequality would not be satisfied.

By following these steps, we can definitively determine whether a given point satisfies a particular inequality. This method provides a simple and reliable way to check if a point lies within the solution set defined by the inequality. In the next section, we will apply this method to the specific point (1, 8) and the inequality y ≤ 2x + 6 to reach a conclusion.

Applying the Method to (1, 8) and y ≤ 2x + 6

Now that we have established the method for verifying a point in an inequality, let's apply it to our specific problem: determining whether the point (1, 8) satisfies the inequality y ≤ 2x + 6. We will follow the steps outlined in the previous section to reach a conclusion.

  1. Identify the Inequality: The inequality we are working with is y ≤ 2x + 6. This inequality defines a region on the coordinate plane consisting of all points where the y-coordinate is less than or equal to 2 times the x-coordinate plus 6.
  2. Identify the Point: The point we want to test is (1, 8). This point has an x-coordinate of 1 and a y-coordinate of 8.
  3. Substitute the Coordinates: We substitute the x-coordinate (1) for x and the y-coordinate (8) for y in the inequality y ≤ 2x + 6. This gives us 8 ≤ 2(1) + 6.
  4. Simplify and Evaluate: Next, we simplify the expression on both sides of the inequality. We have 8 ≤ 2(1) + 6. Multiplying 2 by 1 gives us 8 ≤ 2 + 6. Adding 2 and 6 results in 8 ≤ 8.
  5. Determine if the Inequality Holds True: The resulting statement is 8 ≤ 8. Since 8 is indeed less than or equal to 8, the inequality holds true. This means that the point (1, 8) satisfies the inequality y ≤ 2x + 6.

Therefore, we can confidently conclude that the point (1, 8) is part of the solution set for the inequality y ≤ 2x + 6. This point lies on or below the line defined by the equation y = 2x + 6. In the following sections, we will further explore the implications of this result and discuss the graphical representation of the inequality.

Graphical Representation of the Inequality

A powerful way to visualize inequalities and their solution sets is through graphical representation. The inequality y ≤ 2x + 6 can be represented graphically on the coordinate plane, providing a clear picture of the region that satisfies the inequality. To graph this inequality, we first graph the corresponding equation, y = 2x + 6, which is a straight line.

The equation y = 2x + 6 is in slope-intercept form, where 2 is the slope and 6 is the y-intercept. This means the line crosses the y-axis at the point (0, 6) and rises 2 units for every 1 unit it moves to the right. We can plot a few points on this line, such as (0, 6), (1, 8), and (-1, 4), and then draw a straight line through these points.

Since our inequality is y ≤ 2x + 6, we are interested in all the points where the y-coordinate is less than or equal to 2 times the x-coordinate plus 6. This corresponds to the region on or below the line y = 2x + 6. We represent this region by shading the area below the line. Because the inequality includes the 'equal to' condition (≤), the line itself is part of the solution set, and we draw it as a solid line.

If the inequality were y < 2x + 6, we would draw a dashed line to indicate that the points on the line are not included in the solution set. The shaded region would still be below the line, but the line itself would be excluded.

Now, let's consider the point (1, 8) in the context of this graphical representation. As we found earlier, the point (1, 8) satisfies the inequality y ≤ 2x + 6. When we plot this point on the coordinate plane, we see that it lies exactly on the line y = 2x + 6. This visually confirms our algebraic result, demonstrating that the point (1, 8) is part of the solution set.

The graphical representation provides a valuable visual aid for understanding inequalities and their solutions. It allows us to see the entire region that satisfies the inequality and to quickly determine whether a given point is part of the solution set. In the next section, we will summarize our findings and discuss the significance of our analysis.

Conclusion

In this article, we have explored the process of determining whether a point satisfies an inequality, focusing specifically on the point (1, 8) and the inequality y ≤ 2x + 6. We began by establishing the fundamental concepts of inequalities and their solution sets. We then outlined a step-by-step method for verifying whether a point satisfies an inequality, which involves substituting the coordinates of the point into the inequality and evaluating the resulting statement.

Applying this method to the point (1, 8) and the inequality y ≤ 2x + 6, we found that the inequality holds true. This means that the point (1, 8) is part of the solution set for the inequality. We further reinforced this result by graphically representing the inequality on the coordinate plane. The graph showed that the point (1, 8) lies on the line y = 2x + 6, which is part of the solution region defined by the inequality y ≤ 2x + 6.

This exploration has demonstrated the importance of understanding inequalities and their solutions in mathematics. The ability to verify whether a point satisfies an inequality is a fundamental skill that has applications in various fields, including economics, engineering, and computer science. By combining algebraic methods with graphical representations, we can gain a comprehensive understanding of inequalities and their solution sets.

The example of the point (1, 8) and the inequality y ≤ 2x + 6 serves as a concrete illustration of these concepts. It highlights the process of substituting coordinates, simplifying expressions, and interpreting the results. Moreover, it showcases the power of graphical representations in visualizing mathematical relationships and confirming algebraic findings. This understanding forms a solid foundation for tackling more complex problems involving inequalities and their applications.