Solving Inequalities Step-by-Step Guide
In the realm of mathematics, inequalities play a pivotal role in defining relationships between values that are not necessarily equal. Understanding how to solve inequalities is a fundamental skill, with applications spanning various fields, from economics to engineering. This article provides a detailed walkthrough of solving the inequality 2 3/5 < b - 8/15, offering a clear and concise explanation of each step involved. Let's embark on this journey to master the art of solving inequalities.
Understanding the Basics of Inequalities
Before diving into the specifics of the problem at hand, it's essential to grasp the core concepts of inequalities. Unlike equations, which assert the equality of two expressions, inequalities express a relationship of greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). The inequality 2 3/5 < b - 8/15 signifies that the value of the expression 2 3/5 is strictly less than the value of b - 8/15. Our goal is to isolate the variable 'b' on one side of the inequality to determine the range of values that satisfy this condition.
The process of solving inequalities closely mirrors that of solving equations, with one crucial distinction: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. This rule stems from the nature of the number line, where negative numbers decrease in value as their absolute value increases. With this foundational knowledge in place, we're ready to tackle the given inequality.
Step-by-Step Solution to 2 3/5 < b - 8/15
1. Convert Mixed Numbers to Improper Fractions
The first step in solving the inequality 2 3/5 < b - 8/15 is to convert the mixed number 2 3/5 into an improper fraction. An improper fraction has a numerator that is greater than or equal to its denominator. To convert a mixed number to an improper fraction, we multiply the whole number part (2) by the denominator (5) and add the numerator (3). This result becomes the new numerator, and the denominator remains the same. Thus, 2 3/5 becomes (2 * 5 + 3) / 5 = 13/5. The inequality now reads:
13/5 < b - 8/15
Converting mixed numbers to improper fractions simplifies subsequent arithmetic operations, making it easier to manipulate the inequality and isolate the variable.
2. Isolate the Variable 'b'
Our objective is to isolate the variable 'b' on one side of the inequality. To achieve this, we need to eliminate the term -8/15 from the right side. We can accomplish this by adding 8/15 to both sides of the inequality. This operation maintains the balance of the inequality, ensuring that the relationship between the two sides remains valid. Adding 8/15 to both sides, we get:
13/5 + 8/15 < b - 8/15 + 8/15
Simplifying the right side, -8/15 + 8/15 cancels out, leaving us with 'b' isolated. The inequality now becomes:
13/5 + 8/15 < b
3. Find a Common Denominator
Before we can add the fractions 13/5 and 8/15, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators 5 and 15. The LCM of 5 and 15 is 15. To express 13/5 with a denominator of 15, we multiply both the numerator and denominator by 3:
(13 * 3) / (5 * 3) = 39/15
Now, we can rewrite the inequality with the common denominator:
39/15 + 8/15 < b
Finding a common denominator is a crucial step in adding or subtracting fractions, as it ensures that we are dealing with comparable units.
4. Add the Fractions
With a common denominator in place, we can now add the fractions on the left side of the inequality. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same:
(39 + 8) / 15 < b
This simplifies to:
47/15 < b
5. Convert Improper Fraction to Mixed Number (Optional)
While the inequality 47/15 < b is a perfectly valid solution, it's often helpful to express the improper fraction 47/15 as a mixed number. To do this, we divide the numerator (47) by the denominator (15). The quotient (3) becomes the whole number part of the mixed number, the remainder (2) becomes the numerator, and the denominator (15) remains the same. Thus, 47/15 is equivalent to 3 2/15. The inequality can now be written as:
3 2/15 < b
6. Interpret the Solution
The inequality 3 2/15 < b states that 'b' is greater than 3 2/15. This means that any value of 'b' that is strictly larger than 3 2/15 will satisfy the original inequality 2 3/5 < b - 8/15. We can represent this solution graphically on a number line, where an open circle at 3 2/15 indicates that this value is not included in the solution set, and an arrow extending to the right signifies that all values greater than 3 2/15 are solutions.
Choosing the Correct Answer
Now, let's examine the given options to identify the correct solution:
A. b < 2 1/15 B. b > 2 1/15 C. b < 3 2/15 D. Discussion category :mathematics
Comparing our solution, 3 2/15 < b, with the options, we find that option B, b > 2 1/15, is not the correct answer because it does not match our derived solution. Option C, b < 3 2/15, is also incorrect as it contradicts our solution which states that b is greater than 3 2/15. The correct answer is the one that accurately reflects our solution, which is b > 3 2/15. However, this option is not directly listed. We need to re-evaluate our steps and options to ensure accuracy.
Let's revisit the step where we converted 47/15 to a mixed number. We correctly found it to be 3 2/15. Therefore, the inequality 47/15 < b is equivalent to 3 2/15 < b, which means b > 3 2/15. None of the provided options exactly match this. There might be a slight error in the provided options. However, if we need to choose the closest correct option, it would be B. b > 2 1/15, as it's the only option with the correct inequality direction (greater than). It's crucial to double-check the original problem and options for any potential errors.
Common Mistakes and How to Avoid Them
Solving inequalities, like any mathematical process, is susceptible to errors if not approached with careful attention to detail. One common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This can lead to an incorrect solution set. To avoid this, always double-check the sign of the multiplier or divisor before proceeding.
Another frequent error is making arithmetic mistakes when adding, subtracting, multiplying, or dividing fractions. Ensure you find a common denominator before adding or subtracting fractions, and carefully perform the multiplication and division operations. It's often helpful to write out each step clearly to minimize the chance of errors.
Finally, misinterpreting the solution is a common pitfall. Remember that an inequality represents a range of values, not just a single value. Pay close attention to the direction of the inequality sign and the meaning of the solution in the context of the original problem.
Conclusion
Solving inequalities is a fundamental skill in mathematics with widespread applications. By understanding the basic principles, following a step-by-step approach, and avoiding common mistakes, you can confidently tackle a wide range of inequality problems. In this article, we've meticulously solved the inequality 2 3/5 < b - 8/15, illustrating each step with clarity and precision. We encourage you to practice solving various inequalities to solidify your understanding and enhance your problem-solving abilities. Remember, the key to success in mathematics is consistent practice and a willingness to learn from your mistakes. So, embrace the challenge, hone your skills, and unlock the power of inequalities!
Solving Inequalities Step-by-Step Guide to 2 3/5 < b - 8/15
