Solving 5.25 - B ≥ 6.5 A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving the inequality 5.25 - b ≥ 6.5. Inequalities, a fundamental concept in mathematics, represent relationships where two values are not necessarily equal. Unlike equations, which have a single solution or a finite set of solutions, inequalities often have a range of solutions. Understanding how to manipulate and solve inequalities is crucial for various mathematical applications, from algebra to calculus and beyond. This guide aims to provide a step-by-step approach to solving this particular inequality, ensuring clarity and comprehension for learners of all levels.

Before we begin, let’s clarify some key terms. An inequality is a mathematical statement that compares two expressions using inequality symbols such as greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). Solving an inequality means finding the set of all values that satisfy the inequality. In our case, we want to find all values of b that make the statement 5.25 - b ≥ 6.5 true. The process involves isolating the variable b on one side of the inequality, much like solving an equation, but with an important distinction: multiplying or dividing by a negative number reverses the direction of the inequality sign. This is a critical rule that we will apply in our solution.

This article will not only provide the solution to the inequality but also offer a detailed explanation of each step involved. We will start by outlining the initial problem and then proceed to isolate the variable b. We will discuss the properties of inequalities that allow us to perform operations on both sides without changing the solution set. Additionally, we will address the crucial step of dealing with the negative sign associated with b, which necessitates flipping the inequality sign. Finally, we will present the solution in both inequality and interval notation, ensuring a comprehensive understanding of the result. Through this detailed guide, we aim to equip you with the skills and knowledge to confidently tackle similar inequality problems in the future. By the end of this article, you will have a solid understanding of how to solve inequalities, interpret the solutions, and apply these concepts in broader mathematical contexts. We encourage you to follow along, work through the steps, and practice similar problems to reinforce your learning. This hands-on approach will undoubtedly enhance your proficiency in algebra and your overall mathematical aptitude.

Step-by-Step Solution

To solve the inequality 5.25 - b ≥ 6.5, we need to isolate the variable b on one side of the inequality. This involves a series of algebraic manipulations that preserve the inequality relationship. Our goal is to transform the inequality into a form where b is by itself on one side, allowing us to clearly see the range of values that satisfy the inequality. The process is similar to solving equations, but with a critical difference: we must remember to flip the inequality sign when multiplying or dividing by a negative number.

Step 1: Isolate the Term with 'b'

Our first step is to isolate the term containing b, which in this case is -b. To do this, we need to eliminate the 5.25 from the left side of the inequality. We can accomplish this by subtracting 5.25 from both sides of the inequality. This operation maintains the balance of the inequality, ensuring that the relationship between the two sides remains valid. The original inequality is:

5.25 - b ≥ 6.5

Subtracting 5.25 from both sides, we get:

5. 25 - b - 5.25 ≥ 6.5 - 5.25

Simplifying this, we have:

-b ≥ 1.25

Now, we have successfully isolated the term -b on the left side. This is a crucial step because it brings us closer to isolating b itself. However, we are not quite there yet, as we have -b rather than b. The next step will address this issue and provide us with the solution for b.

Step 2: Solve for 'b'

Now that we have -b ≥ 1.25, we need to solve for b. The presence of the negative sign in front of b requires an additional step. To eliminate the negative sign, we can multiply both sides of the inequality by -1. However, this is where the critical rule for inequalities comes into play: when we multiply or divide an inequality by a negative number, we must reverse the direction of the inequality sign. This is essential to maintain the validity of the inequality.

Multiplying both sides by -1, we get:

(-1) * (-b) ≤ (-1) * 1.25

Notice that the ≥ sign has been flipped to ≤. This is the key step that distinguishes solving inequalities from solving equations. Simplifying the expression, we have:

b ≤ -1.25

This is our solution. It tells us that b is less than or equal to -1.25. This means any value of b that is -1.25 or smaller will satisfy the original inequality. We have successfully solved the inequality by isolating b and applying the rule for multiplying by a negative number.

Step 3: Express the Solution

We have found that the solution to the inequality is b ≤ -1.25. This can be expressed in several ways, each offering a slightly different perspective on the solution set. It's important to be comfortable with these different representations to fully understand the solution and communicate it effectively.

Inequality Notation

The most direct way to express the solution is in inequality notation, which we have already done: b ≤ -1.25. This notation clearly states that b can be any value that is less than or equal to -1.25. It’s concise and easy to understand, making it a common way to represent the solution to inequalities.

Interval Notation

Interval notation is another way to represent the solution set. It uses intervals to describe the range of values that satisfy the inequality. For b ≤ -1.25, the interval notation is (-∞, -1.25]. Let's break this down:

• (-∞ indicates that the solution extends to negative infinity. This means there is no lower bound on the values of b.
• -1.25 is the upper bound of the interval. Since the inequality includes “equal to,” we use a square bracket [ to indicate that -1.25 is included in the solution set.
• The parenthesis ) next to -∞ indicates that infinity is not a number and cannot be included in the interval. We always use parentheses with infinity.

So, the interval notation (-∞, -1.25] represents all real numbers less than or equal to -1.25. This notation is particularly useful when dealing with more complex inequalities or systems of inequalities.

Graphical Representation

Visualizing the solution on a number line can also provide a clear understanding. To represent b ≤ -1.25 graphically, we draw a number line and locate -1.25 on it. Since the inequality includes “equal to,” we use a closed circle (or a filled-in dot) on -1.25 to indicate that it is part of the solution. Then, we shade the portion of the number line to the left of -1.25 to represent all values less than -1.25.

This graphical representation provides a visual confirmation of our solution. The shaded region clearly shows the range of values that b can take while satisfying the inequality.

In summary, the solution to the inequality 5.25 - b ≥ 6.5 is b ≤ -1.25. We can express this solution in inequality notation, interval notation as (-∞, -1.25], and graphically on a number line. Understanding these different representations allows for a more complete comprehension of the solution.

Verification of the Solution

After solving an inequality, it is crucial to verify the solution to ensure its accuracy. Verification involves plugging values from the solution set back into the original inequality to confirm that they indeed satisfy the inequality. This step helps catch any errors made during the solving process and provides confidence in the correctness of the solution. For the inequality 5.25 - b ≥ 6.5, where the solution is b ≤ -1.25, we will perform this verification by testing a few values.

Testing Values

To verify our solution, we will choose three values: one that is equal to -1.25, one that is less than -1.25, and one that is greater than -1.25. This approach will give us a comprehensive check of our solution set. The values we will test are: -1.25, -2, and 0.

Test Value 1: b = -1.25

Substitute b = -1.25 into the original inequality:

5. 25 - (-1.25) ≥ 6.5

Simplifying the left side, we get:

5. 25 + 1.25 ≥ 6.5

6. 5 ≥ 6.5

This statement is true, as 6.5 is equal to 6.5. This confirms that -1.25 is part of the solution set, which aligns with our solution b ≤ -1.25.

Test Value 2: b = -2

Substitute b = -2 into the original inequality:

5. 25 - (-2) ≥ 6.5

Simplifying the left side, we get:

5. 25 + 2 ≥ 6.5

6. 25 ≥ 6.5

This statement is also true, as 7.25 is greater than 6.5. This confirms that a value less than -1.25 satisfies the inequality, which is consistent with our solution b ≤ -1.25.

Test Value 3: b = 0

Substitute b = 0 into the original inequality:

5. 25 - 0 ≥ 6.5

Simplifying the left side, we get:

5. 25 ≥ 6.5

This statement is false, as 5.25 is not greater than or equal to 6.5. This confirms that a value greater than -1.25 does not satisfy the inequality, which further validates our solution b ≤ -1.25.

Conclusion of Verification

By testing these three values, we have verified that our solution b ≤ -1.25 is correct. The values that are less than or equal to -1.25 satisfy the original inequality, while the value greater than -1.25 does not. This verification process underscores the importance of checking solutions to ensure accuracy and provides a solid understanding of the solution set. It also reinforces the concept that the solution to an inequality is a range of values, not just a single value as in the case of an equation. This comprehensive approach to solving inequalities, including verification, is a valuable skill in mathematics and problem-solving.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes that students often make. Recognizing these pitfalls can help you avoid errors and ensure accurate solutions. In this section, we will highlight some of the most frequent mistakes and provide guidance on how to prevent them.

Forgetting to Flip the Inequality Sign

One of the most critical rules when dealing with inequalities is to flip the inequality sign when multiplying or dividing both sides by a negative number. Forgetting this rule is a common mistake that leads to an incorrect solution. The sign must be reversed because multiplying or dividing by a negative number changes the direction of the inequality.

For example, consider the step where we had -b ≥ 1.25. To solve for b, we needed to multiply both sides by -1. If we forget to flip the inequality sign, we might incorrectly write b ≥ -1.25, which is the opposite of the correct solution b ≤ -1.25. To avoid this mistake, always double-check whether you are multiplying or dividing by a negative number and, if so, make sure to reverse the inequality sign.

Incorrectly Applying Operations

Just like with equations, it's essential to apply operations correctly to both sides of the inequality to maintain balance. Mistakes can occur when adding, subtracting, multiplying, or dividing terms. For instance, if you add a number to one side but forget to add it to the other side, the inequality will no longer be valid.

To prevent these errors, it's helpful to write out each step clearly and perform the same operation on both sides. Double-checking your work can also help catch any arithmetic mistakes or incorrect applications of operations. Precision in algebraic manipulation is key to accurately solve inequalities.

Misinterpreting the Solution Set

Once you have solved an inequality, it's important to correctly interpret the solution set. This includes understanding what the inequality notation, interval notation, and graphical representation mean. A common mistake is to misinterpret the direction of the inequality or to include or exclude the endpoint incorrectly.

For example, if the solution is b ≤ -1.25, it means b can be any value less than or equal to -1.25. A mistake would be to think it means b is greater than -1.25. Similarly, in interval notation, (-∞, -1.25] means that -1.25 is included in the solution, while (-∞, -1.25) would mean it is excluded. Understanding these nuances is crucial for accurately representing and interpreting the solution.

Not Verifying the Solution

Failing to verify the solution is another common mistake. As demonstrated earlier, verification is a critical step in ensuring the accuracy of your solution. By plugging values from the solution set back into the original inequality, you can confirm whether they satisfy the inequality. Skipping this step can lead to overlooking errors and accepting an incorrect solution. Always take the time to test your solution with a few values to gain confidence in its correctness. By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and proficiency in solving inequalities. Careful attention to detail, clear writing of steps, and thorough verification are essential practices for success.

Conclusion

In this comprehensive guide, we have explored the process of solving the inequality 5.25 - b ≥ 6.5. We began by understanding the fundamental concepts of inequalities and the importance of isolating the variable to find the solution set. Through a step-by-step approach, we demonstrated how to manipulate the inequality while adhering to the crucial rule of flipping the inequality sign when multiplying or dividing by a negative number. This rule is paramount in preserving the validity of the inequality and arriving at the correct solution.

The solution we found, b ≤ -1.25, was then expressed in various notations, including inequality notation, interval notation, and graphical representation. This multifaceted approach allows for a thorough understanding of the solution set, catering to different learning styles and preferences. We emphasized the significance of interpreting the solution correctly, ensuring that the range of values satisfying the inequality is clearly understood.

Verification played a pivotal role in our guide, highlighting the importance of confirming the accuracy of the solution. By testing values from the solution set in the original inequality, we demonstrated how to validate the answer and gain confidence in its correctness. This step is invaluable in catching any potential errors and reinforcing the understanding of the solution.

Furthermore, we addressed common mistakes that students often encounter when solving inequalities. By identifying these pitfalls, such as forgetting to flip the inequality sign or misinterpreting the solution set, we aimed to equip learners with the knowledge to avoid these errors and improve their problem-solving skills. Clear communication of steps, careful attention to detail, and consistent verification were emphasized as key practices for success.

In conclusion, mastering the skill of solving inequalities is essential in mathematics, with applications extending to various fields. This guide has provided a robust framework for understanding and solving inequalities, focusing on clarity, accuracy, and thoroughness. By following the steps outlined, practicing regularly, and being mindful of common mistakes, you can confidently tackle inequality problems and enhance your mathematical proficiency. The journey through algebra and beyond requires a solid foundation in these fundamental concepts, and we hope this guide has provided you with the tools and insights needed to succeed.