Checking If Point (4,-2) Satisfies Inequality 3x + 2y ≥ 6

Jul 15, 2025 by ADMIN 58 views

Determining if a Point Satisfies the Inequality $3x + 2y

ge 6$ : A Comprehensive Guide

In the realm of mathematics, inequalities play a crucial role in defining regions and constraints. Understanding how to determine if a given point satisfies an inequality is a fundamental skill, particularly in areas like linear programming and optimization. This article delves into the process of verifying whether a point, specifically $(4, -2)$ , satisfies the inequality $3x + 2y ge 6$ . We will explore the underlying concepts, provide a step-by-step solution, and discuss the broader implications of this type of problem. This comprehensive guide aims to equip you with the knowledge and confidence to tackle similar problems effectively.

Understanding Inequalities

Before diving into the specific problem, it's essential to grasp the concept of inequalities. Unlike equations that represent equality between two expressions, inequalities describe a range of possible values. The inequality $3x + 2y ge 6$ indicates that the expression $3x + 2y$ is greater than or equal to 6. This means any pair of values for $x$ and $y$ that make this statement true will satisfy the inequality. Inequalities are often used to represent constraints or limitations in real-world scenarios, such as resource allocation, production capacity, or budget limitations. The graphical representation of an inequality is a region on the coordinate plane, bounded by a line that represents the corresponding equation (in this case, $3x + 2y = 6$ ). Understanding the nature of inequalities is crucial for solving various mathematical problems and for their applications in diverse fields.

Key Takeaways about Inequalities:

Definition: Inequalities express a range of possible values, not just a single solution.
Symbols: Common inequality symbols include >, <, ge, and le.
Graphical Representation: Inequalities are represented by regions on the coordinate plane.
Applications: Used to model constraints and limitations in real-world problems.

Step-by-Step Solution: Verifying the Point (4, -2)

To determine if the point $(4, -2)$ satisfies the inequality $3x + 2y ge 6$ , we need to substitute the coordinates of the point into the inequality and check if the resulting statement is true. This process involves replacing $x$ with 4 and $y$ with -2 in the inequality. Let's break down the steps:

Substitute the Values: Replace $x$ with 4 and $y$ with -2 in the inequality $3x + 2y ge 6$ . This gives us:

$3(4) + 2(-2) ge 6$
Simplify the Expression: Perform the multiplication and addition:

$12 - 4 ge 6$

$8 ge 6$
Evaluate the Statement: Check if the resulting statement is true. In this case, $8 ge 6$ is indeed a true statement since 8 is greater than 6.
Conclusion: Since the statement is true, the point $(4, -2)$ satisfies the inequality $3x + 2y ge 6$ . This means the point lies within the region defined by the inequality on the coordinate plane.

This methodical approach ensures that we accurately determine whether a point satisfies a given inequality. The process is straightforward but requires careful attention to detail, especially when dealing with negative numbers or more complex expressions. By following these steps, you can confidently verify any point against an inequality.

Graphical Interpretation

Visualizing the inequality and the point on a graph can provide a deeper understanding of the solution. The inequality $3x + 2y ge 6$ represents a region on the coordinate plane. To graph this inequality, we first graph the corresponding equation, $3x + 2y = 6$ . This is a straight line that can be plotted by finding two points that satisfy the equation. For example, when $x = 0$ , $y = 3$ , and when $y = 0$ , $x = 2$ . So, the line passes through the points $(0, 3)$ and $(2, 0)$ .

Since the inequality is $3x + 2y ge 6$ , we are interested in the region where the values are greater than or equal to 6. This region is on one side of the line. To determine which side, we can test a point that is not on the line, such as the origin $(0, 0)$ . Substituting $(0, 0)$ into the inequality, we get $3(0) + 2(0) ge 6$ , which simplifies to $0 ge 6$ . This is a false statement, so the region that satisfies the inequality is the side of the line that does not contain the origin.

Now, if we plot the point $(4, -2)$ on the same graph, we will see that it lies within the shaded region representing the inequality $3x + 2y ge 6$ . This visual confirmation reinforces our algebraic solution, demonstrating that the point indeed satisfies the inequality. Graphical interpretation is a powerful tool for understanding inequalities and their solutions, especially in more complex scenarios involving multiple inequalities and constraints.

Broader Implications and Applications

Understanding how to verify if a point satisfies an inequality has significant applications in various fields, particularly in linear programming and optimization problems. Linear programming involves finding the optimal solution (maximum or minimum) of a linear objective function, subject to a set of linear constraints, which are often expressed as inequalities. These constraints define a feasible region, and the optimal solution must lie within this region. Therefore, verifying whether a point satisfies the inequalities is crucial for identifying feasible solutions.

In real-world scenarios, linear programming is used to solve problems related to resource allocation, production planning, transportation logistics, and financial portfolio optimization. For example, a company might use linear programming to determine the optimal production mix of different products, given constraints on raw materials, labor, and production capacity. Each constraint can be expressed as an inequality, and the solution must satisfy all these inequalities.

Furthermore, the concept extends to areas like computer graphics, where inequalities are used to define shapes and regions. In machine learning, support vector machines (SVMs) use inequalities to define decision boundaries that separate different classes of data points. Therefore, mastering the ability to verify points against inequalities is a valuable skill that extends far beyond basic algebra.

Common Mistakes and How to Avoid Them

When working with inequalities, it's essential to be aware of common mistakes to ensure accurate solutions. One frequent error is incorrectly substituting the values of the point into the inequality. Double-check your substitutions to make sure you've replaced $x$ and $y$ with the correct coordinates.

Another common mistake occurs when simplifying the expression after substitution. Pay close attention to the order of operations (PEMDAS/BODMAS) and be careful with negative signs. An incorrect simplification can lead to an inaccurate result and a wrong conclusion about whether the point satisfies the inequality.

Furthermore, when dealing with inequalities involving multiplication or division by a negative number, remember to flip the inequality sign. Forgetting this rule is a common error that can change the direction of the inequality and lead to incorrect solutions.

To avoid these mistakes, it's helpful to show your work step by step, double-check each step, and practice regularly. If possible, use graphing tools to visualize the inequality and the point, which can provide a visual confirmation of your algebraic solution. By being mindful of these common pitfalls and taking precautions, you can improve your accuracy and confidence in working with inequalities.

Practice Problems

To solidify your understanding, let's consider a few practice problems:

Does the point $(-1, 5)$ satisfy the inequality $2x - y < 4$ ?
Does the point $(3, -2)$ satisfy the inequality $x + 4y le -5$ ?
Does the point $(0, 0)$ satisfy the inequality $-3x + 2y > 1$ ?

For each problem, follow the steps outlined earlier: substitute the coordinates into the inequality, simplify the expression, and evaluate the resulting statement. Compare your answers with the solutions below:

$2(-1) - 5 < 4 ightarrow -2 - 5 < 4 ightarrow -7 < 4$ . Yes, the point $(-1, 5)$ satisfies the inequality.
$3 + 4(-2) le -5 ightarrow 3 - 8 le -5 ightarrow -5 le -5$ . Yes, the point $(3, -2)$ satisfies the inequality.
$-3(0) + 2(0) > 1 ightarrow 0 > 1$ . No, the point $(0, 0)$ does not satisfy the inequality.

Working through these practice problems will reinforce your understanding and help you develop the skills needed to solve similar problems confidently.

Conclusion

In conclusion, determining whether a point satisfies an inequality is a fundamental skill with wide-ranging applications. By following a systematic approach, including substitution, simplification, and evaluation, you can confidently verify points against inequalities. The process is not only crucial for solving mathematical problems but also for understanding and applying mathematical concepts in real-world scenarios, such as linear programming and optimization. By avoiding common mistakes and practicing regularly, you can master this skill and enhance your problem-solving abilities in mathematics and beyond. This guide has provided a comprehensive overview of the process, from the basic understanding of inequalities to their graphical interpretation and practical applications. With this knowledge, you are well-equipped to tackle inequality problems with confidence and accuracy.