Finding Solutions For Inequalities A Detailed Explanation For Y ≤ 4x + 5

Introduction to Linear Inequalities

In the realm of mathematics, linear inequalities play a crucial role in describing relationships where one expression is either less than, greater than, less than or equal to, or greater than or equal to another. Unlike linear equations, which have a single solution or a finite set of solutions, linear inequalities often have an infinite number of solutions. This article delves into the process of identifying solutions to the linear inequality y ≤ 4x + 5. We will explore the concept of solutions to inequalities, the graphical representation of inequalities, and how to determine whether a given point satisfies the inequality. Understanding these concepts is fundamental for various applications in mathematics, economics, and other fields where constraints and limitations are involved.

At its core, a linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). In the context of two-variable inequalities, we are dealing with inequalities involving two variables, typically denoted as x and y. The solutions to such inequalities are not single points but rather regions in the coordinate plane. These regions are bounded by lines, which are the graphical representations of the corresponding linear equations. The inequality y ≤ 4x + 5 represents all the points (x, y) in the coordinate plane that satisfy the condition that the y-coordinate is less than or equal to 4 times the x-coordinate plus 5. This includes all points on the line y = 4x + 5 and all points below this line. Identifying which points are solutions to this inequality involves substituting the coordinates of the points into the inequality and checking if the resulting statement is true. This process allows us to determine whether a given point lies within the solution region of the inequality.

To fully grasp the concept of solving linear inequalities, it's essential to understand the graphical representation. The inequality y ≤ 4x + 5 corresponds to a region in the Cartesian plane. The boundary of this region is the line y = 4x + 5. This line divides the plane into two halves, one where y is less than or equal to 4x + 5 and the other where y is greater than 4x + 5. The solution to the inequality y ≤ 4x + 5 includes all the points on the line y = 4x + 5 and all the points below this line. This region is often shaded to visually represent the solution set. When we test a point to see if it is a solution, we are essentially checking if the point lies within this shaded region. For instance, if we substitute the coordinates of a point into the inequality and find that the inequality holds true, then the point is a solution and lies within the shaded region. Conversely, if the inequality does not hold true, the point is not a solution and lies outside the shaded region. This graphical interpretation provides a clear and intuitive way to understand the solutions of linear inequalities and how they relate to the coordinate plane.

Analyzing the Inequality y ≤ 4x + 5

To determine which point is a solution to the inequality y ≤ 4x + 5, we need to test each given point by substituting its coordinates into the inequality. This involves replacing 'x' and 'y' in the inequality with the x-coordinate and y-coordinate of the point, respectively. If the resulting statement is true, then the point is a solution; otherwise, it is not. This process is a fundamental method for verifying solutions to inequalities and is widely used in various mathematical contexts. By systematically testing each point, we can identify which ones satisfy the given inequality and thus belong to the solution set. The inequality y ≤ 4x + 5 represents a region in the coordinate plane that includes the line y = 4x + 5 and all points below it. Therefore, any point that lies on or below this line is a solution to the inequality.

Let's consider each of the given points and apply this method. For point A (0, 10), we substitute x = 0 and y = 10 into the inequality. This gives us 10 ≤ 4(0) + 5, which simplifies to 10 ≤ 5. This statement is false, meaning that point A (0, 10) is not a solution to the inequality. Next, for point B (0, -2), we substitute x = 0 and y = -2 into the inequality. This yields -2 ≤ 4(0) + 5, which simplifies to -2 ≤ 5. This statement is true, indicating that point B (0, -2) is a solution to the inequality. Moving on to point C (-4, 0), we substitute x = -4 and y = 0 into the inequality, resulting in 0 ≤ 4(-4) + 5, which simplifies to 0 ≤ -16 + 5, or 0 ≤ -11. This statement is false, so point C (-4, 0) is not a solution. Finally, for point D (-6, 4), we substitute x = -6 and y = 4 into the inequality, giving us 4 ≤ 4(-6) + 5, which simplifies to 4 ≤ -24 + 5, or 4 ≤ -19. This statement is also false, meaning that point D (-6, 4) is not a solution. By systematically testing each point, we have determined that only point B (0, -2) satisfies the inequality y ≤ 4x + 5.

This method of substituting coordinates into the inequality is a straightforward and reliable way to check if a point is a solution. It relies on the fundamental understanding of what an inequality represents and how points in the coordinate plane relate to algebraic expressions. When the inequality holds true after substitution, it means that the point lies within the solution region of the inequality. This solution region can be visualized graphically as the area bounded by the line corresponding to the equation formed by replacing the inequality sign with an equals sign. In the case of y ≤ 4x + 5, the line is y = 4x + 5, and the solution region includes all points on and below this line. By testing each point, we are essentially checking if it falls within this region. This approach is not only useful for solving mathematical problems but also for understanding the broader concept of inequalities and their applications in various fields.

Step-by-Step Verification of Each Option

To meticulously determine the solution to the inequality y ≤ 4x + 5, we will now perform a step-by-step verification of each provided option. This process involves substituting the x and y coordinates of each point into the inequality and evaluating whether the resulting statement is true or false. This method is crucial for understanding how points in the coordinate plane relate to inequalities and for accurately identifying solutions. By carefully examining each option, we can pinpoint the point that satisfies the given condition, thereby reinforcing our understanding of linear inequalities and their solutions.

Option A: (0, 10)

For option A, the point is (0, 10). We substitute x = 0 and y = 10 into the inequality y ≤ 4x + 5. This gives us:10 ≤ 4(0) + 5. Simplifying the right side, we get:10 ≤ 0 + 5, which further simplifies to:10 ≤ 5. This statement is false. Therefore, the point (0, 10) is not a solution to the inequality y ≤ 4x + 5. This means that the point lies outside the solution region of the inequality, which is the area on or below the line y = 4x + 5. In the coordinate plane, the point (0, 10) is located relatively high, far above the line, and thus does not satisfy the condition of the inequality.

Option B: (0, -2)

For option B, the point is (0, -2). Substituting x = 0 and y = -2 into the inequality y ≤ 4x + 5, we get:-2 ≤ 4(0) + 5. Simplifying the right side, we have:-2 ≤ 0 + 5, which simplifies to:-2 ≤ 5. This statement is true. Therefore, the point (0, -2) is a solution to the inequality y ≤ 4x + 5. This point lies within the solution region, either on the line y = 4x + 5 or below it. In the coordinate plane, the point (0, -2) is located below the x-axis and is indeed below the line, thus satisfying the inequality.

Option C: (-4, 0)

For option C, the point is (-4, 0). Substituting x = -4 and y = 0 into the inequality y ≤ 4x + 5, we get:0 ≤ 4(-4) + 5. Simplifying the right side, we have:0 ≤ -16 + 5, which simplifies to:0 ≤ -11. This statement is false. Therefore, the point (-4, 0) is not a solution to the inequality y ≤ 4x + 5. This point is not within the solution region, indicating it lies above the line y = 4x + 5. In the coordinate plane, the point (-4, 0) is on the x-axis but is positioned such that it does not satisfy the condition of the inequality.

Option D: (-6, 4)

For option D, the point is (-6, 4). Substituting x = -6 and y = 4 into the inequality y ≤ 4x + 5, we get:4 ≤ 4(-6) + 5. Simplifying the right side, we have:4 ≤ -24 + 5, which simplifies to:4 ≤ -19. This statement is false. Therefore, the point (-6, 4) is not a solution to the inequality y ≤ 4x + 5. This point also lies outside the solution region, above the line y = 4x + 5. In the coordinate plane, the point (-6, 4) is in the second quadrant, and its position is such that it does not meet the criteria defined by the inequality.

Conclusion: Identifying the Correct Solution

After systematically verifying each option against the inequality y ≤ 4x + 5, we have conclusively determined that only one point satisfies the given condition. The process involved substituting the coordinates of each point into the inequality and checking if the resulting statement was true. This method is a fundamental technique in algebra for determining whether a point lies within the solution region of an inequality. By understanding this process, we can confidently solve similar problems and gain a deeper understanding of the relationship between algebraic inequalities and their graphical representations.

Through our step-by-step analysis, we found that:

• Point A (0, 10) did not satisfy the inequality, as 10 ≤ 5 is a false statement.
• Point B (0, -2) satisfied the inequality, as -2 ≤ 5 is a true statement.
• Point C (-4, 0) did not satisfy the inequality, as 0 ≤ -11 is a false statement.
• Point D (-6, 4) did not satisfy the inequality, as 4 ≤ -19 is a false statement.

Therefore, the correct solution is point B (0, -2). This point is the only one among the given options that lies within the solution region of the inequality y ≤ 4x + 5. This region includes all points on the line y = 4x + 5 and all points below it. The graphical representation of the inequality further clarifies this concept, as the solution region can be visualized as a shaded area in the coordinate plane.

In conclusion, the ability to identify solutions to inequalities is a crucial skill in mathematics. It not only helps in solving problems related to linear inequalities but also lays the foundation for understanding more complex mathematical concepts. The method of substituting coordinates and verifying the resulting statement is a reliable and straightforward approach that can be applied to various types of inequalities. By mastering this technique, students can enhance their problem-solving abilities and gain a deeper appreciation for the elegance and power of mathematics.