Finding Integer Solutions Inequality -3 ≤ 2x - 1 ≤ 5

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#Introduction

In the realm of mathematics, inequalities play a crucial role in defining ranges and boundaries for variables. Integer solutions to inequalities are of particular interest, as they represent specific whole number values that satisfy the given conditions. This article delves into the process of finding the number of integer solutions that satisfy the inequality -3 ≤ 2x - 1 ≤ 5. We will explore the steps involved in solving this compound inequality and identifying the integers that fall within the solution set. Understanding how to solve such inequalities is fundamental in various mathematical contexts, including algebra, calculus, and discrete mathematics. This exploration will not only provide a solution to the specific problem but also enhance your understanding of inequality solving techniques in general.

Understanding the Inequality

At its core, the inequality -3 ≤ 2x - 1 ≤ 5 represents a compound inequality, essentially two inequalities combined into one expression. This notation signifies that the expression 2x - 1 is simultaneously greater than or equal to -3 and less than or equal to 5. To dissect this, we can break it down into two separate inequalities: -3 ≤ 2x - 1 and 2x - 1 ≤ 5. Solving each of these individually will give us a range for x, and the overlap between these ranges will be the solution set for the original compound inequality. This is a fundamental concept in algebra, allowing us to deal with complex conditions by addressing them in manageable parts. Visualizing this on a number line can also be helpful. Imagine a number line where we are looking for the values of 2x - 1 that fall between -3 and 5, inclusive. This sets the stage for our algebraic manipulations to isolate x and determine the precise integer solutions. The comprehension of compound inequalities is a cornerstone of mathematical problem-solving, enabling us to tackle problems involving constraints and limitations. Mastering this concept opens doors to more advanced topics, such as linear programming and optimization problems. So, let's embark on a step-by-step journey to unravel the solution to this fascinating inequality.

Solving the Inequality

To solve the inequality -3 ≤ 2x - 1 ≤ 5, our primary goal is to isolate x, the variable for which we seek integer solutions. We achieve this by performing algebraic manipulations on all parts of the inequality, ensuring that we maintain the balance and validity of the expression. Our first step involves adding 1 to all parts of the inequality. This eliminates the -1 term on the left side of the expression involving x, bringing us closer to isolating x. Adding 1 to all parts of -3 ≤ 2x - 1 ≤ 5 gives us -3 + 1 ≤ 2x - 1 + 1 ≤ 5 + 1, which simplifies to -2 ≤ 2x ≤ 6. Next, we need to get rid of the coefficient 2 that is multiplying x. To do this, we divide all parts of the inequality by 2. Remember, when dividing or multiplying an inequality by a positive number, the direction of the inequality signs remains unchanged. Dividing all parts of -2 ≤ 2x ≤ 6 by 2 yields -2/2 ≤ 2x/2 ≤ 6/2, which simplifies to -1 ≤ x ≤ 3. Now we have successfully isolated x, and the inequality -1 ≤ x ≤ 3 represents the range of all possible solutions for x. This range includes all real numbers between -1 and 3, including -1 and 3 themselves. However, our focus is on integer solutions, which we will identify in the next section. The process of isolating the variable through algebraic manipulation is a core skill in algebra and is essential for solving a wide variety of equations and inequalities.

Identifying Integer Solutions

Having solved the inequality -3 ≤ 2x - 1 ≤ 5 and found the range -1 ≤ x ≤ 3, our next task is to identify the integer solutions within this range. Remember, integers are whole numbers (without any fractional or decimal parts), which can be positive, negative, or zero. Looking at the inequality -1 ≤ x ≤ 3, we need to find all the integers that are greater than or equal to -1 and less than or equal to 3. Starting from -1, we can list the integers in ascending order: -1, 0, 1, 2, and 3. These are the only whole numbers that fall within the specified range. Therefore, the integer solutions to the inequality are -1, 0, 1, 2, and 3. To confirm that these values are indeed solutions, we can substitute each of them back into the original inequality -3 ≤ 2x - 1 ≤ 5 and check if the inequality holds true. For example, if we substitute x = 0, we get -3 ≤ 2(0) - 1 ≤ 5, which simplifies to -3 ≤ -1 ≤ 5, which is true. Similarly, we can verify the other solutions. Identifying integer solutions from a range is a crucial skill in various mathematical contexts, particularly in discrete mathematics and number theory. It allows us to focus on specific, countable values that satisfy a given condition. In this case, we have successfully identified the five integer solutions that make the inequality -3 ≤ 2x - 1 ≤ 5 true.

Counting the Solutions

Now that we have identified the integer solutions to the inequality -3 ≤ 2x - 1 ≤ 5 as -1, 0, 1, 2, and 3, the final step is to count how many such solutions exist. This is a straightforward process once we have the complete list of integers that satisfy the inequality. By simply counting the elements in the set {-1, 0, 1, 2, 3}, we find that there are five integer solutions. This means that there are five whole number values for x that make the inequality -3 ≤ 2x - 1 ≤ 5 true. This result provides a concise answer to the original problem. In many mathematical problems, determining the number of solutions is as important as finding the solutions themselves. This is particularly relevant in areas such as combinatorics and probability, where we often need to count the number of ways something can occur. In this case, we have determined that there are five integers that satisfy the given inequality, providing a complete and satisfying resolution to the problem. The ability to count solutions is a fundamental skill in mathematics and has wide-ranging applications in various fields.

Verification of Solutions

To ensure the accuracy of our solution, it is crucial to verify that each identified integer indeed satisfies the original inequality -3 ≤ 2x - 1 ≤ 5. This process involves substituting each integer value back into the inequality and checking if the resulting statement holds true. Let's take each integer solution and perform the verification:

  • For x = -1: -3 ≤ 2(-1) - 1 ≤ 5 simplifies to -3 ≤ -3 ≤ 5, which is true.
  • For x = 0: -3 ≤ 2(0) - 1 ≤ 5 simplifies to -3 ≤ -1 ≤ 5, which is true.
  • For x = 1: -3 ≤ 2(1) - 1 ≤ 5 simplifies to -3 ≤ 1 ≤ 5, which is true.
  • For x = 2: -3 ≤ 2(2) - 1 ≤ 5 simplifies to -3 ≤ 3 ≤ 5, which is true.
  • For x = 3: -3 ≤ 2(3) - 1 ≤ 5 simplifies to -3 ≤ 5 ≤ 5, which is true.

As we can see, each of the identified integer solutions (-1, 0, 1, 2, and 3) satisfies the inequality -3 ≤ 2x - 1 ≤ 5. This verification process not only confirms the correctness of our solution but also reinforces our understanding of the inequality and its solution set. Verification is a vital step in problem-solving, especially in mathematics. It helps to catch any potential errors and ensures that our final answer is accurate and reliable. By systematically verifying each solution, we can have confidence in our result and the process we used to obtain it.

Conclusion

In conclusion, we have successfully determined the number of integer solutions that satisfy the inequality -3 ≤ 2x - 1 ≤ 5. By systematically solving the compound inequality, we found the range of possible values for x to be -1 ≤ x ≤ 3. We then identified the integers within this range as -1, 0, 1, 2, and 3. Finally, we counted these integers and found that there are five integer solutions in total. Each of these solutions was verified to ensure its validity. This exercise demonstrates the importance of algebraic manipulation, understanding inequalities, and identifying integer solutions. The process of solving inequalities and finding integer solutions is a fundamental skill in mathematics with applications in various fields. By mastering these techniques, we can approach a wide range of problems with confidence and accuracy. The ability to solve inequalities is not only valuable in academic settings but also in real-world scenarios where constraints and limitations need to be considered. This exploration has not only provided a solution to the specific problem but also reinforced the core concepts and techniques involved in solving inequalities.