Solving Systems Of Inequalities Finding The Ordered Pair Solution

In mathematics, solving a system of inequalities involves identifying the region on a coordinate plane where all the inequalities are simultaneously satisfied. This means finding the set of ordered pairs (x, y) that make all the given inequalities true. This article delves into the process of determining which ordered pair satisfies a given system of inequalities, providing a step-by-step approach and clear explanations to enhance understanding.

Understanding Systems of Inequalities

Before tackling the problem, let's grasp the concept of a system of inequalities. A system of inequalities is a set of two or more inequalities involving the same variables. The solution to a system of inequalities is the set of all ordered pairs that satisfy all the inequalities in the system. Graphically, this solution is represented by the region where the shaded areas of all inequalities overlap. Understanding the nature of inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) is crucial. Each symbol dictates the boundary line's inclusion or exclusion (dashed vs. solid line) and the region to be shaded.

When faced with the problem of finding an ordered pair that makes both inequalities true, we are essentially looking for a point (x, y) that lies within the overlapping shaded region of the inequalities when graphed. This point must satisfy each inequality individually, making it a solution to the system.

Problem Statement

Our specific problem presents us with two inequalities:

1. $y < 3x - 1$
2. $y ≥ -x + 4$

And four potential ordered pairs:

A. (4, 0) B. (1, 2) C. (0, 4) D. (2, 1)

The task is to determine which, if any, of these ordered pairs satisfies both inequalities. To achieve this, we will substitute the x and y values of each ordered pair into the inequalities and check if the resulting statements are true.

Step-by-Step Solution

Let's methodically test each ordered pair against the inequalities:

A. (4, 0)

• Inequality 1: $y < 3x - 1$ Substitute x = 4 and y = 0: $0 < 3(4) - 1$ $0 < 12 - 1$ $0 < 11$ (True)
• Inequality 2: $y ≥ -x + 4$ Substitute x = 4 and y = 0: $0 ≥ -(4) + 4$ $0 ≥ -4 + 4$ $0 ≥ 0$ (True)

Since (4, 0) satisfies both inequalities, it is a potential solution.

B. (1, 2)

• Inequality 1: $y < 3x - 1$ Substitute x = 1 and y = 2: $2 < 3(1) - 1$ $2 < 3 - 1$ $2 < 2$ (False)

Since (1, 2) fails to satisfy the first inequality, there's no need to test the second one. It is not a solution.

C. (0, 4)

• Inequality 1: $y < 3x - 1$ Substitute x = 0 and y = 4: $4 < 3(0) - 1$ $4 < 0 - 1$ $4 < -1$ (False)

Similarly, (0, 4) does not satisfy the first inequality and is therefore not a solution.

D. (2, 1)

• Inequality 1: $y < 3x - 1$ Substitute x = 2 and y = 1: $1 < 3(2) - 1$ $1 < 6 - 1$ $1 < 5$ (True)
• Inequality 2: $y ≥ -x + 4$ Substitute x = 2 and y = 1: $1 ≥ -(2) + 4$ $1 ≥ -2 + 4$ $1 ≥ 2$ (False)

(2, 1) satisfies the first inequality but not the second, so it is not a solution.

Conclusion

After testing all the ordered pairs, we find that only (4, 0) satisfies both inequalities in the system. Therefore, (4, 0) is the solution to the problem. This systematic approach of substituting the x and y values into each inequality is a reliable method for solving such problems.

The Importance of Understanding Inequalities

Understanding inequalities is fundamental in various mathematical and real-world applications. Inequalities are used to describe situations where quantities are not equal, providing a range of possible values rather than a single, definitive answer. In fields like economics, inequalities can model budget constraints or resource allocation. In engineering, they can define tolerance levels or safety margins. The ability to solve and interpret inequalities is, therefore, a valuable skill.

Graphing inequalities provides a visual representation of the solution set, making it easier to understand the range of possible values. The shaded region on the graph represents all the ordered pairs that satisfy the inequality. When dealing with a system of inequalities, the overlapping shaded region represents the solution set for the entire system. This graphical approach is particularly helpful in understanding the relationships between different inequalities and identifying feasible regions in optimization problems.

Tips for Solving Systems of Inequalities

Here are some helpful tips to keep in mind when solving systems of inequalities:

1. Understand the Inequality Symbols: Be clear on the meaning of each symbol (<, >, ≤, ≥) and how it affects the solution set.
2. Isolate the Variable: If necessary, rearrange the inequalities to isolate the variable you are solving for. This makes it easier to substitute values and check solutions.
3. Test Points Systematically: When given a set of ordered pairs, test each pair in each inequality. This methodical approach ensures you don't miss any potential solutions.
4. Graph the Inequalities: Graphing the inequalities can provide a visual representation of the solution set and help you understand the range of possible values. The overlapping shaded region represents the solution to the system.
5. Check Boundary Lines: Pay attention to whether the boundary lines are included in the solution set (solid lines for ≤ and ≥) or excluded (dashed lines for < and >).
6. Use Technology: Utilize graphing calculators or online tools to graph inequalities and visualize the solution set. This can be especially helpful for complex systems of inequalities.

Real-World Applications of Systems of Inequalities

The concepts of systems of inequalities extend beyond the classroom and into various real-world scenarios. These applications highlight the practical significance of understanding and solving inequalities.

1. Budgeting and Finance: Inequalities can represent budget constraints, showing the different combinations of goods or services a person can afford given their income. For example, an inequality can represent the total cost of items being less than or equal to the available budget.
2. Resource Allocation: In business and management, inequalities are used to optimize resource allocation. For instance, a company might use inequalities to determine the optimal mix of products to manufacture given constraints on raw materials, labor, and production capacity.
3. Nutrition and Diet Planning: Inequalities can help in planning a balanced diet by setting minimum and maximum limits for nutrient intake. Dieticians use inequalities to ensure that individuals consume adequate amounts of vitamins, minerals, and macronutrients while staying within caloric limits.
4. Engineering Design: Engineers use inequalities to define tolerance levels and safety margins in designs. For example, an inequality might specify the acceptable range of stress a structural component can withstand.
5. Optimization Problems: Many optimization problems in various fields, such as logistics, operations research, and computer science, involve solving systems of inequalities to find the best possible solution within given constraints.

By mastering the skills to solve systems of inequalities, students and professionals alike can tackle these real-world problems effectively. The ability to analyze and interpret inequalities is a valuable asset in decision-making and problem-solving across various domains.

In conclusion, finding the ordered pair that makes both inequalities true involves a systematic approach of testing each option against the inequalities. Understanding the fundamental concepts of inequalities, graphical representation, and real-world applications enhances the problem-solving skills in mathematics and beyond.