Solving Inequalities System For Positive Integers A And B
In the realm of mathematics, inequalities play a crucial role in defining relationships between variables and setting boundaries for their possible values. When faced with a system of inequalities, our goal is to find the set of solutions that satisfy all the inequalities simultaneously. This article delves into a problem involving two positive integers, a and b, subject to certain conditions expressed as inequalities. We will explore how to translate these conditions into a system of inequalities and identify the correct representation. This problem not only tests our understanding of inequalities but also our ability to interpret and apply mathematical concepts in a real-world context. The core challenge lies in accurately capturing the given relationships—the sum and the difference of the integers—using mathematical notation. Furthermore, we must consider the constraint that b is the greater integer, which adds another layer to the problem-solving process. By dissecting the problem step by step, we will arrive at the system of inequalities that precisely describes the possible values of a and b. This exercise is fundamental in various fields, including optimization problems, linear programming, and decision-making processes where constraints and conditions need to be mathematically modeled. Mastering the art of translating word problems into mathematical inequalities is a valuable skill that extends beyond the classroom and into practical applications.
Problem Statement
The core of the problem lies in deciphering the relationships between two positive integers, a and b, and translating them into a mathematical form. To recap, we have two positive integers, a and b, with the following conditions:
- Their sum is at least 30.
- Their difference is at least 10, and b is the greater integer.
The challenge is to determine which system of inequalities accurately represents these conditions. We are presented with two options:
* Option 1:
* *a* + *b* ≥ 30
* *b* ≥ *a* + 10
* Option 2:
* *a* + *b* ≥ 30
* *b* ≤ *a* – 10
To solve this, we must carefully analyze each condition and how it translates into an inequality. The phrase "at least" signifies a greater than or equal to relationship (≥). The first condition, "The sum of a and b is at least 30," directly translates to the inequality a + b ≥ 30. This part is straightforward and common to both options, which simplifies our initial assessment. However, the second condition is where the crux of the problem lies. The statement "The difference of the two integers is at least 10, and b is the greater integer" requires a more nuanced approach. We need to express the difference between b and a (since b is greater) as being greater than or equal to 10. This is where the two options diverge, presenting a critical point of analysis. Our task is to scrutinize each option in light of this condition, ensuring that the chosen inequality accurately reflects the given relationship between a and b. By methodically breaking down the problem and focusing on the precise meaning of each statement, we can confidently identify the correct system of inequalities.
Translating the Conditions into Inequalities
To accurately represent the given conditions as inequalities, we need to break down each statement and convert it into mathematical notation. The first condition states that "The sum of two positive integers, a and b, is at least 30." The phrase "at least" indicates a minimum value, meaning the sum can be 30 or greater. This translates directly into the inequality:
a + b ≥ 30
This inequality establishes a lower bound for the sum of a and b. Any pair of positive integers that satisfies this inequality will have a sum of 30 or more. Now, let's tackle the second condition: "The difference of the two integers is at least 10, and b is the greater integer." This statement has two key components: the difference between the integers and the fact that b is the larger integer. Since b is the greater integer, the difference is calculated as b - a. The phrase "at least 10" means this difference must be 10 or greater. Therefore, we can express this condition as:
b - a ≥ 10
However, the options provided in the problem express this relationship differently. To match the given format, we can rearrange the inequality by adding a to both sides:
b ≥ a + 10
This form highlights that b is greater than or equal to a plus 10. It reinforces the condition that b is significantly larger than a. The rearrangement not only helps in comparing with the given options but also provides a clearer understanding of the relationship between a and b. By carefully translating each condition into its mathematical equivalent, we lay the foundation for identifying the correct system of inequalities. The precision in this translation is paramount to ensuring an accurate solution. We have now successfully converted both conditions into inequalities, paving the way for the next step in our problem-solving process.
Analyzing the Options
Having translated the problem's conditions into inequalities, we now face the critical task of analyzing the provided options to identify the correct system. Let's revisit the two options:
- Option 1:
- a + b ≥ 30
- b ≥ a + 10
- Option 2:
- a + b ≥ 30
- b ≤ a – 10
The first inequality, a + b ≥ 30, is common to both options and accurately reflects the condition that the sum of a and b is at least 30. This means our focus should be on the second inequality in each option, as this is where they diverge. Recall that we translated the second condition, "The difference of the two integers is at least 10, and b is the greater integer," into the inequality:
b ≥ a + 10
This inequality states that b is greater than or equal to a plus 10. Now, let's compare this with the second inequalities in each option. Option 1 includes the inequality b ≥ a + 10, which perfectly matches our translated condition. This suggests that Option 1 is a strong candidate for the correct system of inequalities. On the other hand, Option 2 includes the inequality b ≤ a – 10. This inequality states that b is less than or equal to a minus 10, which contradicts the condition that b is the greater integer and their difference is at least 10. This discrepancy immediately raises a red flag for Option 2. To further solidify our analysis, we can consider a few examples. If we take a = 10 and b = 20, the sum a + b is 30, satisfying the first inequality. In Option 1, b ≥ a + 10 translates to 20 ≥ 10 + 10, which is true. However, in Option 2, b ≤ a - 10 translates to 20 ≤ 10 - 10, which is false. This example further supports the correctness of Option 1 and the incorrectness of Option 2. By carefully comparing the translated conditions with the given options and testing with examples, we can confidently pinpoint the system of inequalities that accurately represents the problem.
Identifying the Correct System of Inequalities
Based on our analysis, we can now definitively identify the correct system of inequalities. We established that the conditions given in the problem translate to the following inequalities:
- a + b ≥ 30
- b ≥ a + 10
When we compare these inequalities with the provided options:
- Option 1:
- a + b ≥ 30
- b ≥ a + 10
- Option 2:
- a + b ≥ 30
- b ≤ a – 10
It becomes clear that Option 1 perfectly matches the inequalities we derived from the problem statement. The first inequality, a + b ≥ 30, correctly represents the condition that the sum of a and b is at least 30. The second inequality, b ≥ a + 10, accurately captures the condition that the difference between b and a is at least 10, given that b is the greater integer. Option 2, on the other hand, contains the inequality b ≤ a – 10, which contradicts the given condition that b is the greater integer and their difference is at least 10. This inequality implies that b is less than a, which is the opposite of what the problem states. Therefore, Option 2 is incorrect. To further validate our conclusion, we can consider a few examples that satisfy Option 1. For instance, if a = 10 and b = 20, both inequalities hold true: 10 + 20 ≥ 30 and 20 ≥ 10 + 10. Conversely, these values would not satisfy Option 2. By methodically translating the conditions, analyzing the options, and validating with examples, we have confidently identified the correct system of inequalities. The ability to accurately represent real-world scenarios using mathematical inequalities is a fundamental skill with wide-ranging applications in various fields.
In summary, the problem presented a scenario involving two positive integers, a and b, with specific conditions regarding their sum and difference. The task was to identify the system of inequalities that accurately represents these conditions. Through a step-by-step approach, we successfully translated the given statements into mathematical inequalities, analyzed the provided options, and identified the correct system. We began by carefully dissecting each condition. The statement "The sum of two positive integers, a and b, is at least 30" was directly translated into the inequality a + b ≥ 30. The more nuanced condition, "The difference of the two integers is at least 10, and b is the greater integer," was expressed as b ≥ a + 10. This translation required understanding that "at least" implies a greater than or equal to relationship and recognizing that the difference should be calculated as b - a since b is the larger integer. Next, we analyzed the provided options, focusing on the second inequality in each system, as the first inequality was common to both. By comparing our translated inequalities with the options, we immediately recognized that Option 1 (a + b ≥ 30 and b ≥ a + 10) perfectly matched our derived inequalities. Option 2 (a + b ≥ 30 and b ≤ a – 10), on the other hand, contained an inequality that contradicted the given condition that b is the greater integer. To further validate our conclusion, we used examples. We demonstrated that values satisfying Option 1 also satisfied the original conditions, while values that might seem plausible could fail under Option 2's constraints. This methodical approach—translating, analyzing, and validating—is crucial in solving mathematical problems involving inequalities. The ability to accurately represent real-world scenarios using mathematical notation is a valuable skill that extends beyond the classroom and into various practical applications. Therefore, the correct system of inequalities that represents the values of a and b is:
- a + b ≥ 30
- b ≥ a + 10
This problem underscores the importance of precision in mathematical interpretation and the power of translating verbal conditions into symbolic form. By mastering these skills, we can effectively tackle a wide range of problems involving inequalities and mathematical relationships.
