Solving (5/6)x - (1/3) > 1(1/3) A Step-by-Step Guide
Introduction: Understanding Inequalities and the Target Problem
In the realm of mathematics, inequalities play a crucial role in defining ranges and boundaries for solutions. Unlike equations that pinpoint specific values, inequalities describe a set of values that satisfy a given condition. This article delves into the process of solving a specific inequality: (5/6)x - (1/3) > 1(1/3). We will meticulously break down each step, ensuring a clear understanding of the underlying principles and arriving at the correct solution. This exploration will not only provide the answer but also enhance your comprehension of inequality manipulation and problem-solving strategies. Before diving into the solution, it's essential to grasp the fundamental concepts of inequalities. Inequalities use symbols like '>', '<', '≥', and '≤' to express the relative order of two values. The greater than symbol (>) indicates that one value is larger than the other, while the less than symbol (<) signifies the opposite. The greater than or equal to symbol (≥) and the less than or equal to symbol (≤) include the possibility of equality. When solving inequalities, our goal is to isolate the variable, much like we do with equations. However, a critical difference arises when multiplying or dividing by a negative number: the direction of the inequality sign must be reversed. This stems from the nature of the number line and how negative numbers interact with inequalities. For instance, if we have -2 < 4 and multiply both sides by -1, we get 2 > -4, demonstrating the need to flip the sign to maintain the inequality's truth. In the given problem, (5/6)x - (1/3) > 1(1/3), we aim to find the range of values for 'x' that make the inequality hold true. This involves a series of algebraic manipulations to isolate 'x' on one side of the inequality. Let's embark on this step-by-step journey to unveil the solution, ensuring a solid grasp of each operation and its rationale. This comprehensive approach will empower you to tackle similar problems with confidence and precision. Solving inequalities is a fundamental skill in algebra and is applied in various fields, including optimization problems, economics, and computer science.
Step 1: Converting Mixed Numbers to Improper Fractions
The first step in solving the inequality (5/6)x - (1/3) > 1(1/3) is to convert the mixed number, 1(1/3), into an improper fraction. This conversion simplifies the subsequent calculations and allows us to work with fractions more efficiently. A mixed number combines a whole number and a fraction, while 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 by the denominator of the fraction and then add the numerator. This result becomes the new numerator, and the denominator remains the same. In our case, we have 1(1/3). The whole number is 1, the numerator is 1, and the denominator is 3. Multiplying the whole number (1) by the denominator (3) gives us 3. Adding the numerator (1) to this result yields 4. Therefore, the improper fraction equivalent of 1(1/3) is 4/3. Now, we can rewrite the original inequality with the improper fraction: (5/6)x - (1/3) > 4/3. This transformation sets the stage for the next steps in solving for 'x'. Working with improper fractions offers several advantages. It eliminates the need to keep track of whole numbers separately, making algebraic manipulations smoother and less prone to errors. Furthermore, improper fractions are easier to work with when performing operations like addition, subtraction, multiplication, and division. By converting the mixed number to an improper fraction, we ensure that all terms in the inequality are expressed in a consistent format, paving the way for a more straightforward solution process. This seemingly simple step is crucial for maintaining accuracy and clarity throughout the problem-solving journey. Understanding how to convert mixed numbers to improper fractions is a fundamental skill in arithmetic and is essential for tackling more complex mathematical problems involving fractions. This skill not only simplifies calculations but also enhances your overall mathematical fluency. By mastering this conversion, you'll be well-equipped to handle various types of equations and inequalities with greater ease and confidence.
Step 2: Eliminating Fractions by Finding the Least Common Denominator (LCD)
To further simplify the inequality (5/6)x - (1/3) > 4/3, the next logical step is to eliminate the fractions. This is achieved by finding the least common denominator (LCD) of all the fractions present in the inequality and then multiplying both sides of the inequality by the LCD. The LCD is the smallest positive integer that is divisible by all the denominators in the fractions. In our case, the denominators are 6, 3, and 3. To find the LCD, we can list the multiples of each denominator and identify the smallest multiple that appears in all lists. Multiples of 6: 6, 12, 18, 24, ... Multiples of 3: 3, 6, 9, 12, ... The smallest common multiple is 6. Therefore, the LCD of 6, 3, and 3 is 6. Now, we multiply both sides of the inequality by 6. This multiplication will effectively clear the fractions, making the inequality easier to manipulate. Multiplying both sides of (5/6)x - (1/3) > 4/3 by 6, we get: 6 * [(5/6)x - (1/3)] > 6 * (4/3) Applying the distributive property on the left side, we have: 6 * (5/6)x - 6 * (1/3) > 6 * (4/3) Simplifying each term: (6/6) * 5x - (6/3) > (6/3) * 4 This simplifies to: 5x - 2 > 8 By multiplying both sides of the inequality by the LCD, we have successfully transformed the inequality into one without fractions. This significantly reduces the complexity of the problem and allows us to proceed with the remaining steps more smoothly. Eliminating fractions is a common technique in solving equations and inequalities involving fractions. It streamlines the process and minimizes the chances of making errors. The LCD method is particularly effective because it ensures that the resulting equation or inequality has integer coefficients, which are easier to work with. This technique is a cornerstone of algebraic manipulation and is frequently used in various mathematical contexts. Understanding and applying the LCD method is crucial for mastering fraction-related problems and building a solid foundation in algebra.
Step 3: Isolating the Variable Term
Having eliminated the fractions, our inequality now stands as 5x - 2 > 8. The next step in solving for 'x' is to isolate the variable term, which in this case is '5x'. To achieve this, we need to eliminate the constant term, '-2', from the left side of the inequality. This can be done by adding 2 to both sides of the inequality. Adding the same value to both sides of an inequality maintains the inequality's balance, just as it does in equations. This principle is a fundamental property of inequalities and is crucial for performing valid algebraic manipulations. Adding 2 to both sides of 5x - 2 > 8, we get: 5x - 2 + 2 > 8 + 2 This simplifies to: 5x > 10 By adding 2 to both sides, we have successfully isolated the variable term '5x' on the left side of the inequality. The inequality now reads '5x > 10', which is much closer to the solution. Isolating the variable term is a key step in solving any algebraic equation or inequality. It brings us closer to expressing the variable in terms of constants, which ultimately reveals the solution set. This step often involves adding or subtracting constants from both sides of the inequality, ensuring that the variable term is the only term remaining on one side. The process of isolating the variable term highlights the importance of maintaining balance in inequalities. Any operation performed on one side must be mirrored on the other side to preserve the truth of the inequality. This principle is a direct consequence of the properties of inequalities and is essential for arriving at the correct solution. Mastering the technique of isolating the variable term is crucial for solving a wide range of algebraic problems. It is a fundamental skill that forms the basis for more advanced mathematical concepts and applications. This step-by-step approach ensures that each manipulation is valid and contributes to the overall goal of finding the solution set for the inequality. The ability to confidently isolate variables is a hallmark of strong algebraic proficiency.
Step 4: Solving for x
With the inequality now simplified to 5x > 10, the final step in finding the solution is to isolate 'x' completely. This involves dividing both sides of the inequality by the coefficient of 'x', which in this case is 5. Dividing both sides of an inequality by a positive number maintains the direction of the inequality sign. However, it is crucial to remember that if we were to divide by a negative number, we would need to reverse the inequality sign. This is a fundamental rule in inequality manipulation and is essential for obtaining the correct solution. Dividing both sides of 5x > 10 by 5, we get: (5x) / 5 > 10 / 5 This simplifies to: x > 2 Therefore, the solution to the inequality (5/6)x - (1/3) > 1(1/3) is x > 2. This means that any value of 'x' greater than 2 will satisfy the original inequality. The solution set can be represented graphically on a number line, with an open circle at 2 (since x is strictly greater than 2) and an arrow extending to the right, indicating all values greater than 2. Solving for 'x' involves isolating the variable on one side of the inequality, expressing it in terms of constants. This process often entails dividing or multiplying both sides of the inequality by a constant. The key is to perform the same operation on both sides to maintain the balance and ensure the validity of the solution. The final solution, x > 2, provides a clear and concise answer to the original problem. It defines the range of values for 'x' that make the inequality true. This solution is a result of a series of logical and systematic steps, each building upon the previous one. Mastering the process of solving for 'x' is crucial for success in algebra and other mathematical disciplines. It requires a solid understanding of inequality properties and the ability to apply them correctly. The solution to the inequality, x > 2, represents a set of infinite values that satisfy the condition. This highlights a key difference between equations and inequalities: equations typically have a finite number of solutions, while inequalities often have an infinite range of solutions. This concept is fundamental to understanding the nature of inequalities and their applications in various mathematical contexts.
Conclusion: Summarizing the Solution and its Implications
In conclusion, the solution to the inequality (5/6)x - (1/3) > 1(1/3) is x > 2. This solution was derived through a series of steps, each designed to simplify the inequality and isolate the variable 'x'. We began by converting the mixed number to an improper fraction, which allowed us to work with fractions more efficiently. Next, we eliminated the fractions by finding the least common denominator (LCD) and multiplying both sides of the inequality by it. This step transformed the inequality into one with integer coefficients, making it easier to manipulate. We then isolated the variable term by adding 2 to both sides of the inequality. Finally, we solved for 'x' by dividing both sides by 5, arriving at the solution x > 2. This solution indicates that any value of 'x' greater than 2 will satisfy the original inequality. The process of solving this inequality highlights several important principles of algebra. First, it demonstrates the importance of maintaining balance in inequalities. Any operation performed on one side must also be performed on the other side to preserve the truth of the inequality. Second, it underscores the significance of understanding the properties of inequalities, such as the rule that dividing by a negative number reverses the inequality sign. Third, it illustrates the power of systematic problem-solving. By breaking down a complex problem into smaller, manageable steps, we can arrive at the solution with clarity and confidence. The solution x > 2 has various implications. In a practical context, it might represent a minimum requirement or a threshold that must be exceeded. For example, if 'x' represents the number of hours a student needs to study to achieve a certain grade, then the solution x > 2 indicates that the student must study for more than 2 hours. In a mathematical context, the solution can be used to define a region on a number line or in a coordinate plane. The set of all numbers greater than 2 forms an open interval that extends infinitely to the right. This concept is fundamental in calculus and other advanced mathematical fields. Understanding how to solve inequalities is a crucial skill in mathematics and has wide-ranging applications in various disciplines. This article has provided a detailed step-by-step guide to solving a specific inequality, but the principles and techniques discussed can be applied to a wide range of similar problems. By mastering these skills, you can enhance your problem-solving abilities and build a solid foundation for future mathematical endeavors. The journey from the initial inequality to the final solution, x > 2, showcases the elegance and power of algebraic manipulation. Each step is a testament to the logical structure of mathematics and the ability to derive meaningful results from seemingly complex problems. This understanding is key to unlocking further mathematical insights and applications.
