Solve The Following In The Set Of Integers: A) 3 + 2(x + 2) = 5 B) 2(x + 5) + 3 < -9 C) |2x - 1| < 5

Introduction

In the realm of mathematics, solving equations and inequalities within the set of integers is a fundamental skill. This article delves into the process of finding integer solutions for various algebraic expressions. We will explore specific examples, providing a step-by-step approach to solving each problem. Understanding these techniques is crucial for success in algebra and number theory. The ability to manipulate equations and inequalities, while adhering to the constraints of integer solutions, demonstrates a strong grasp of mathematical principles. This guide aims to equip you with the necessary tools and strategies to confidently tackle such problems. Whether you are a student learning the basics or someone looking to refresh your skills, this comprehensive guide will provide valuable insights and practical methods for solving equations and inequalities in the set of integers. We will cover a range of examples, from simple linear equations to more complex inequalities involving absolute values, ensuring a thorough understanding of the subject matter. By the end of this article, you will be well-versed in the techniques required to approach and solve a variety of integer-based mathematical problems. Furthermore, we will emphasize the importance of verifying your solutions to ensure accuracy and avoid common pitfalls. So, let's embark on this mathematical journey and unlock the secrets of solving equations and inequalities within the integer domain. Remember, practice is key, and with each problem you solve, your understanding and confidence will grow. This article is designed to be your go-to resource for mastering the art of integer equation and inequality solving.

a) 3 + 2(x + 2) = 5

To solve the equation 3 + 2(x + 2) = 5 in the set of integers, we need to isolate the variable 'x'. This involves a series of algebraic manipulations, ensuring each step preserves the equality. Our primary goal is to find the value(s) of 'x' that satisfy the equation, keeping in mind that 'x' must be an integer. Let's break down the solution step-by-step to ensure clarity and understanding. First, we simplify the equation by distributing the '2' across the terms inside the parentheses: 3 + 2x + 4 = 5. This step is crucial as it removes the parentheses, making the equation easier to work with. Next, we combine the constant terms on the left side of the equation: 7 + 2x = 5. Now, we want to isolate the term with 'x'. To do this, we subtract '7' from both sides of the equation: 2x = 5 - 7, which simplifies to 2x = -2. Finally, we divide both sides by '2' to solve for 'x': x = -2 / 2, which gives us x = -1. It's essential to verify our solution by substituting x = -1 back into the original equation: 3 + 2(-1 + 2) = 5. Simplifying this, we get 3 + 2(1) = 5, which further simplifies to 3 + 2 = 5. This confirms that 5 = 5, so our solution x = -1 is correct. Since -1 is an integer, it satisfies the requirement that the solution must be within the set of integers. Therefore, the solution to the equation 3 + 2(x + 2) = 5 in the set of integers is x = -1. This step-by-step approach ensures accuracy and helps in understanding the underlying algebraic principles involved in solving equations. By following these steps, you can confidently solve similar equations and verify your solutions to avoid errors.

b) 2(x + 5) + 3 < -9

To solve the inequality 2(x + 5) + 3 < -9 in the set of integers, we follow a similar process to solving equations, but with a crucial difference: we must consider how the inequality sign behaves when multiplying or dividing by negative numbers. Our aim is to isolate the variable 'x' while preserving the inequality. Let's break down the solution into manageable steps. First, distribute the '2' across the terms inside the parentheses: 2x + 10 + 3 < -9. This simplifies the inequality and prepares it for further manipulation. Next, combine the constant terms on the left side: 2x + 13 < -9. Now, we want to isolate the term with 'x'. Subtract '13' from both sides of the inequality: 2x < -9 - 13, which simplifies to 2x < -22. To solve for 'x', divide both sides by '2': x < -11. This inequality tells us that 'x' must be less than -11. However, we are looking for integer solutions, so we need to identify the integers that satisfy this condition. The integers less than -11 are -12, -13, -14, and so on. Therefore, the solution set in the integers is {x ∈ ℤ | x < -11}. This means any integer less than -11 is a valid solution to the inequality. It's important to note that we did not need to flip the inequality sign because we divided by a positive number. If we had divided by a negative number, we would have had to reverse the inequality sign. The solution set represents all integers that satisfy the given inequality. Understanding this concept is crucial for solving inequalities accurately. By following these steps and paying attention to the properties of inequalities, you can confidently solve a wide range of problems in the set of integers.

c) |2x - 1| < 5

Solving the inequality |2x - 1| < 5 in the set of integers requires a slightly different approach due to the absolute value. The absolute value of a number is its distance from zero, so |2x - 1| < 5 means that the expression 2x - 1 is within 5 units of zero. This gives us two separate inequalities to solve. First, we consider the case where the expression inside the absolute value is positive or zero: 2x - 1 < 5. To solve this, we add '1' to both sides: 2x < 6. Then, we divide both sides by '2': x < 3. This gives us one part of the solution. Next, we consider the case where the expression inside the absolute value is negative: -(2x - 1) < 5. This is equivalent to -2x + 1 < 5. Subtract '1' from both sides: -2x < 4. Now, we divide both sides by '-2'. Remember, when dividing by a negative number, we must reverse the inequality sign: x > -2. This gives us the other part of the solution. Combining these two inequalities, we have -2 < x < 3. This means that 'x' must be greater than -2 and less than 3. Since we are looking for integer solutions, we need to identify the integers that fall within this range. The integers that satisfy this condition are -1, 0, 1, and 2. Therefore, the solution set in the integers is {-1, 0, 1, 2}. We can verify these solutions by plugging them back into the original inequality |2x - 1| < 5. For example, if x = -1, |2(-1) - 1| = |-3| = 3, which is less than 5. If x = 0, |2(0) - 1| = |-1| = 1, which is less than 5. If x = 1, |2(1) - 1| = |1| = 1, which is less than 5. If x = 2, |2(2) - 1| = |3| = 3, which is less than 5. All these solutions satisfy the original inequality. This step-by-step approach, considering both positive and negative cases of the absolute value expression, is crucial for solving absolute value inequalities accurately. By understanding these principles, you can confidently tackle a variety of similar problems.

Conclusion

In conclusion, solving equations and inequalities in the set of integers requires a solid understanding of algebraic manipulation and the properties of inequalities. We've explored three distinct examples, each showcasing different techniques and considerations. From simple linear equations to inequalities involving absolute values, we've demonstrated a systematic approach to finding integer solutions. The key takeaways include the importance of simplifying expressions, isolating the variable, and paying attention to the impact of operations on inequality signs. Additionally, we've emphasized the crucial step of verifying solutions to ensure accuracy. By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical problems involving integers. Remember that practice is essential, and each problem you solve will further solidify your understanding and confidence. Whether you're a student preparing for an exam or simply someone interested in mathematics, the skills and knowledge gained from this guide will prove invaluable. The ability to solve equations and inequalities in the set of integers is a fundamental skill that extends beyond the classroom and into various real-world applications. So, continue to practice, explore different types of problems, and challenge yourself to deepen your understanding of this important mathematical concept. With dedication and the right approach, you can excel in solving integer-based mathematical problems and unlock new levels of mathematical proficiency.