Solving 6x - 4 Less Than 8 A Step By Step Guide

When it comes to inequalities, understanding how to solve them is a crucial skill in mathematics. Inequalities, unlike equations, deal with relationships where one side is not necessarily equal to the other. Instead, we use symbols like '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to) to express these relationships. This article will delve into the process of solving the inequality $6x - 4 < 8$, providing a step-by-step explanation and ensuring clarity for learners of all levels. The ability to solve inequalities is fundamental not only in algebra but also in various real-world applications, from determining budget constraints to optimizing resource allocation. We aim to break down the process into manageable steps, making it accessible and understandable for anyone eager to grasp this essential mathematical concept. Let's embark on this journey together and unlock the mysteries of solving inequalities.

The core principle behind solving inequalities is similar to solving equations: we aim to isolate the variable on one side of the inequality. However, there's a critical difference: when we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the number line. For example, while 2 < 3, multiplying both sides by -1 gives -2 > -3. Keeping this rule in mind is essential for accurately solving inequalities. The step-by-step approach we'll use includes simplifying both sides of the inequality, performing operations to isolate the variable term, and finally, isolating the variable itself. Each step builds upon the previous one, ensuring that we maintain the integrity of the inequality throughout the process. Solving inequalities is more than just manipulating symbols; it's about understanding the relationships between quantities and expressing those relationships mathematically. This foundational skill opens doors to more advanced topics in mathematics and provides a valuable tool for problem-solving in various domains.

Understanding the properties of inequalities is also vital. We can add or subtract the same number from both sides of an inequality without changing the solution set. Similarly, we can multiply or divide both sides by the same positive number without affecting the solution. These properties allow us to manipulate inequalities in a way that helps us isolate the variable. Consider the inequality $6x - 4 < 8$. Our goal is to get 'x' by itself on one side. To do this, we'll first address the constant term (-4) on the left side. By adding 4 to both sides, we maintain the balance of the inequality while moving closer to isolating the term with 'x'. This simple yet powerful technique forms the basis for solving a wide range of inequalities. It's not just about following a set of rules; it's about understanding why these rules work and how they help us to find the solution. As we progress through the steps, we'll see how each operation contributes to simplifying the inequality and revealing the range of values for 'x' that satisfy the original condition. Solving inequalities is a journey of logical deduction, and each step brings us closer to the final destination: the solution set.

Step-by-Step Solution of 6x - 4 < 8

1. Isolate the Variable Term

The initial step in solving the inequality $6x - 4 < 8$ involves isolating the term that contains the variable, which in this case is $6x$. To achieve this, we need to eliminate the constant term, $-4$, from the left side of the inequality. We can accomplish this by applying the addition property of inequalities. This property states that adding the same number to both sides of an inequality preserves the inequality's validity. Therefore, we will add $4$ to both sides of the inequality.

By adding $4$ to both sides, we get:

$6x - 4 + 4 < 8 + 4$

This simplifies to:

$6x < 12$

This step is crucial because it brings us closer to isolating the variable $x$. By eliminating the constant term from the left side, we've simplified the inequality and made it easier to proceed with the next steps. The addition property of inequalities is a fundamental tool in solving inequalities, and mastering its application is essential for success in this area of mathematics. It's not just about adding a number; it's about understanding how this operation helps us to simplify the inequality and move towards the solution. The next step will build upon this foundation, bringing us even closer to isolating the variable and finding the solution set.

2. Isolate the Variable

Having successfully isolated the variable term $6x$ on the left side of the inequality, our next objective is to isolate the variable $x$ itself. The inequality we currently have is $6x < 12$. To get $x$ by itself, we need to eliminate the coefficient $6$ that is multiplying it. We can do this by applying the division property of inequalities. This property states that dividing both sides of an inequality by the same positive number preserves the inequality's direction. Since $6$ is a positive number, we can safely divide both sides of the inequality by $6$ without flipping the inequality sign.

Dividing both sides of $6x < 12$ by $6$, we get:

rac{6x}{6} < rac{12}{6}

This simplifies to:

$x < 2$

This step is the culmination of our efforts, as it provides us with the solution to the inequality. We have successfully isolated the variable $x$ and determined the condition it must satisfy. The division property of inequalities is a powerful tool, but it's crucial to remember the caveat about dividing by a negative number. In this case, since we divided by a positive number, we didn't need to worry about flipping the inequality sign. The result, $x < 2$, tells us that any value of $x$ less than $2$ will satisfy the original inequality. This solution set is a range of values, not just a single number, which is a key difference between solving inequalities and solving equations.

3. Interpret the Solution

The solution we've arrived at, $x < 2$, is a mathematical statement that describes a set of values. It tells us that $x$ can be any number that is strictly less than $2$. This means that $2$ itself is not included in the solution set. To fully understand the solution, it's helpful to visualize it. We can represent the solution graphically on a number line. On the number line, we would draw an open circle at $2$ to indicate that $2$ is not included in the solution, and then we would shade the line to the left of $2$ to represent all the values less than $2$. This visual representation provides a clear picture of the solution set and helps to reinforce the concept that inequalities often have a range of solutions rather than a single solution.

The solution $x < 2$ can also be expressed in interval notation. Interval notation is a concise way of representing a set of numbers using intervals. In this case, the solution set includes all numbers from negative infinity up to (but not including) $2$. Therefore, in interval notation, the solution is written as $(-\infty, 2)$. The parenthesis next to $2$ indicates that $2$ is not included in the interval. Understanding interval notation is essential for communicating mathematical solutions effectively. It's a standard way of expressing sets of numbers and is widely used in higher-level mathematics. The combination of the inequality notation ($x < 2$), the number line representation, and the interval notation $(-\infty, 2)$ provides a comprehensive understanding of the solution set and its implications.

Choosing the Correct Option

Having solved the inequality $6x - 4 < 8$, we found that the solution is $x < 2$. Now, we need to match this solution with the options provided. Let's examine the options:

A. $x \leq 2$

B. $x < 2$

C. $x < \frac{4}{6}$

D. $x > 2$

By comparing our solution $x < 2$ with the options, we can see that option B, $x < 2$, perfectly matches our result. This option correctly represents the set of all numbers less than $2$. The other options are incorrect. Option A, $x \leq 2$, includes $2$ in the solution set, which is not part of our solution. Option C, $x < \frac{4}{6}$, represents a smaller range of values than our solution. Option D, $x > 2$, represents values greater than $2$, which is the opposite of our solution. Therefore, the correct answer is option B.

Choosing the correct option is the final step in the problem-solving process. It's essential to carefully compare the solution we've derived with the given options to ensure an accurate answer. This step reinforces the understanding of the solution and its representation in different forms. The ability to match a solution with its correct representation is a crucial skill in mathematics, as it demonstrates a complete understanding of the problem and its solution.

Common Mistakes and How to Avoid Them

Solving inequalities, while conceptually straightforward, can be prone to errors if certain precautions are not taken. Understanding these common pitfalls and learning how to avoid them is crucial for mastering the topic. One of the most frequent mistakes is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. As we discussed earlier, this operation reverses the order of the number line, and failing to account for this will lead to an incorrect solution. For example, if we had an inequality like $-2x < 4$, dividing both sides by $-2$ requires us to flip the inequality sign, resulting in $x > -2$, not $x < -2$. To avoid this mistake, always double-check whether you're multiplying or dividing by a negative number and remember to flip the sign if you are.

Another common mistake is incorrectly applying the order of operations. Just like in equations, we need to follow the correct order of operations (PEMDAS/BODMAS) when simplifying inequalities. This means dealing with parentheses, exponents, multiplication and division, and finally, addition and subtraction, in the correct sequence. Failing to do so can lead to incorrect simplification and ultimately, the wrong solution. For instance, in the inequality $3(x + 2) < 9$, we need to distribute the $3$ before we can isolate the variable. A mistake would be to subtract 2 from both sides first, which would lead to an incorrect result. To prevent this, always take a moment to identify the correct order of operations and apply them systematically.

Finally, a lack of understanding of the solution set can also lead to errors. The solution to an inequality is often a range of values, not just a single number. It's important to understand how to represent this range using inequality notation, number lines, and interval notation. A common mistake is to misinterpret the meaning of the inequality symbols. For example, $x < 2$ means all numbers less than 2, but not including 2 itself, while $x \leq 2$ includes 2 in the solution set. To avoid this, practice interpreting the different inequality symbols and visualizing the solution set on a number line. This will help you to develop a deeper understanding of what the solution represents and prevent errors in your answers.

Conclusion: Mastering Inequalities

In conclusion, solving the inequality $6x - 4 < 8$ involves a systematic approach that includes isolating the variable term, isolating the variable itself, and correctly interpreting the solution. We've walked through each step in detail, highlighting the key principles and properties of inequalities. The solution we found, $x < 2$, represents all numbers less than $2$, and this corresponds to option B in the given choices. Understanding inequalities is not just about following steps; it's about grasping the underlying concepts and applying them correctly.

By mastering the techniques discussed in this article, you'll be well-equipped to tackle a wide range of inequality problems. Remember to pay close attention to the rules, especially when multiplying or dividing by a negative number, and always double-check your work. With practice and a solid understanding of the fundamentals, you can confidently solve inequalities and expand your mathematical abilities.