Solving Inequalities Step-by-Step Guide (3/2)x - 1 < 8x - (7/6)

Jul 10, 2025 by ADMIN 64 views

Solving Inequalities A Comprehensive Guide to Solving the Inequality (3/2)x - 1 < 8x - (7/6)

In the realm of mathematics, inequalities play a crucial role in defining relationships between expressions that are not necessarily equal. Solving inequalities, much like solving equations, involves finding the range of values that satisfy a given condition. This article delves into the process of solving the inequality (3/2)x - 1 < 8x - (7/6), providing a step-by-step guide and comprehensive explanations to enhance your understanding. Inequalities are pervasive in various fields, from economics to physics, and mastering their resolution is essential for problem-solving and decision-making. Understanding how to manipulate and solve inequalities not only strengthens your mathematical foundation but also equips you with the analytical skills necessary to tackle real-world challenges. This article aims to break down the complexity of solving inequalities into manageable steps, ensuring clarity and confidence in your approach. The journey through this mathematical landscape will empower you to not only solve this specific inequality but also apply the learned principles to a wide array of similar problems. So, let’s embark on this enlightening exploration together, unraveling the intricacies and mastering the art of solving inequalities.

Before diving into the solution, it's important to grasp the concept of inequalities. Unlike equations, which assert the equality of two expressions, inequalities express a relationship where two expressions are not necessarily equal. The symbols used in inequalities include: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Inequalities are fundamental in mathematics and are used to describe relationships where exact equality is not present. They provide a range of possible values that satisfy a given condition, making them indispensable tools in various fields such as optimization, economics, and engineering. The ability to interpret and manipulate inequalities is a cornerstone of mathematical proficiency. Understanding the nuances of these symbols and their implications is essential for accurate problem-solving and logical reasoning. In the subsequent sections, we will explore how these concepts apply to solving our specific inequality, ensuring a solid grasp of the underlying principles. Grasping the fundamentals of inequalities is not just about understanding symbols; it's about developing a mindset that appreciates the spectrum of possibilities rather than just the certainty of equality. This perspective is crucial for advanced mathematical studies and real-world applications alike.

1. Eliminate Fractions

To simplify the inequality, our first step is to eliminate the fractions. The least common multiple (LCM) of the denominators (2 and 6) is 6. Multiply both sides of the inequality by 6:

6 * ((3/2)x - 1) < 6 * (8x - (7/6))

This simplifies to:

9x - 6 < 48x - 7

Eliminating fractions is a crucial first step in simplifying inequalities. By multiplying both sides by the least common multiple (LCM) of the denominators, we effectively clear the fractions, making the inequality easier to manipulate. This step transforms the original inequality into a more manageable form, allowing us to proceed with subsequent operations without the burden of fractional coefficients. The process not only simplifies the algebraic expressions but also reduces the likelihood of errors in subsequent calculations. This technique is widely applicable in solving various types of inequalities and equations, demonstrating its significance in algebraic manipulations. Mastering the elimination of fractions is a fundamental skill that enhances one's ability to tackle complex mathematical problems with greater efficiency and accuracy. By clearing the fractions, we pave the way for a smoother solution process, ensuring that the focus remains on the core algebraic manipulations rather than getting bogged down by complex fractions.

2. Isolate the Variable

Next, we want to isolate the variable x. Subtract 9x from both sides:

9x - 6 - 9x < 48x - 7 - 9x

This gives us:

-6 < 39x - 7

Now, add 7 to both sides:

-6 + 7 < 39x - 7 + 7

Which simplifies to:

1 < 39x

Isolating the variable is a critical step in solving any inequality or equation. This process involves rearranging the terms in the inequality so that the variable (in this case, 'x') is on one side and the constants are on the other. By performing the same operations on both sides of the inequality, we maintain the balance and ensure that the solution remains valid. The steps involved in isolating the variable often include adding, subtracting, multiplying, or dividing both sides by appropriate values. This technique is not only fundamental in solving inequalities but also crucial in various algebraic manipulations. Mastery of this skill allows for a systematic approach to problem-solving, enabling one to unravel complex equations and inequalities with confidence. The ability to isolate the variable is a cornerstone of algebraic proficiency, paving the way for a deeper understanding of mathematical concepts and their applications in real-world scenarios. This step-by-step process ensures that we are one step closer to determining the range of values that satisfy the given inequality.

3. Solve for x

Finally, divide both sides by 39:

1/39 < x

This can also be written as:

x > 1/39

Solving for x is the culmination of the inequality-solving process. After isolating the variable, the final step involves performing the necessary operation to determine the range of values that satisfy the inequality. In this case, we divide both sides of the inequality by the coefficient of x, which is 39. This operation effectively isolates x, revealing the solution set. The resulting inequality, x > 1/39, indicates that any value of x greater than 1/39 will satisfy the original inequality. This step is crucial as it provides a clear and concise answer to the problem. Understanding how to solve for x is a fundamental skill in algebra and is essential for tackling various mathematical problems. The ability to accurately determine the solution set not only demonstrates a strong grasp of algebraic principles but also enhances problem-solving capabilities in a wide range of contexts. This final step solidifies the solution and provides a clear understanding of the range of values that make the inequality true.

Therefore, the solution to the inequality (3/2)x - 1 < 8x - (7/6) is x > 1/39. This means any value of x greater than 1/39 will satisfy the original inequality.

The solution x > 1/39 can be visualized on a number line. Draw a number line and mark 1/39. Since x is strictly greater than 1/39, we use an open circle at 1/39 and shade the region to the right, indicating all values greater than 1/39. Visualizing solutions on a number line is a powerful tool for understanding inequalities. It provides a clear and intuitive representation of the range of values that satisfy the inequality. The use of open or closed circles indicates whether the endpoint is included in the solution set, and the shaded region represents all the values that make the inequality true. This visual aid is particularly helpful for students as it bridges the gap between abstract algebraic concepts and concrete representations. The number line visualization not only enhances comprehension but also aids in problem-solving by providing a graphical perspective. This method is widely used in mathematics education to reinforce the understanding of inequalities and their solutions. By visualizing the solution, we gain a deeper insight into the nature of the inequality and its implications.

To verify our solution, we can pick a value greater than 1/39 and substitute it into the original inequality. Let’s choose x = 1:

(3/2)(1) - 1 < 8(1) - (7/6)

3/2 - 1 < 8 - 7/6

1/2 < 41/6

This is true, so our solution is correct. Checking the solution is a crucial step in the process of solving inequalities and equations. By substituting a value from the solution set back into the original inequality, we can verify whether the solution is correct. This step helps to identify any errors that may have occurred during the algebraic manipulations. If the chosen value satisfies the inequality, it provides confidence in the correctness of the solution. However, if the inequality is not satisfied, it indicates that there may be an error in the solution process, prompting a review of the steps taken. Checking the solution is not just a procedural step; it is an integral part of the problem-solving process that reinforces understanding and accuracy. This practice is highly recommended for all mathematical problems as it ensures that the final answer is valid and reliable. By verifying the solution, we solidify our understanding and minimize the chances of errors.

When solving inequalities, there are several common mistakes to avoid:

Forgetting to flip the inequality sign when multiplying or dividing by a negative number. This is a crucial rule and a frequent source of errors.
Incorrectly distributing multiplication over parentheses. Ensure you multiply each term inside the parentheses.
Making arithmetic errors. Double-check your calculations, especially when dealing with fractions or negative numbers.
Misinterpreting the inequality symbols. Understand the difference between <, >, ≤, and ≥.

Avoiding common mistakes is essential for accurate problem-solving in mathematics. One of the most frequent errors in solving inequalities is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. This rule is crucial and must be applied correctly to maintain the validity of the solution. Another common mistake is incorrect distribution of multiplication over parentheses, which can lead to errors in simplifying expressions. Arithmetic errors, especially when dealing with fractions or negative numbers, are also a significant source of mistakes. It is essential to double-check calculations to ensure accuracy. Misinterpreting the inequality symbols, such as <, >, ≤, and ≥, can also lead to incorrect solutions. A clear understanding of these symbols is vital for correctly interpreting and solving inequalities. By being aware of these common pitfalls and taking the necessary precautions, one can significantly improve their accuracy in solving mathematical problems. Avoiding these mistakes not only ensures correct solutions but also enhances overall problem-solving skills.

Solving inequalities is a fundamental skill in mathematics with wide-ranging applications. By following the steps outlined in this article, you can confidently solve the inequality (3/2)x - 1 < 8x - (7/6) and similar problems. Remember to eliminate fractions, isolate the variable, and pay attention to the direction of the inequality sign. With practice, you'll become proficient in solving inequalities and applying this skill to various mathematical contexts. The ability to solve inequalities is not just about finding the correct answer; it's about developing logical reasoning and problem-solving skills that are valuable in many areas of life. Mastering this skill opens doors to more advanced mathematical concepts and real-world applications. So, keep practicing, and embrace the challenge of solving inequalities!