Solving For X In The Inequality 3x + 3 - X + (-7) > 6

Unveiling the Secrets of Solving for x in Linear Inequalities

In the realm of mathematics, solving for x is a fundamental skill that unlocks the doors to understanding equations and inequalities. This article delves into the process of solving for x in a linear inequality, providing a comprehensive guide for students and enthusiasts alike. We will dissect the given inequality, $3x + 3 - x + (-7) > 6$, step by step, revealing the underlying principles and techniques involved. By mastering these concepts, you'll be well-equipped to tackle a wide range of algebraic challenges. The beauty of algebra lies in its ability to represent real-world scenarios with concise mathematical expressions. Inequalities, in particular, allow us to model situations where quantities are not necessarily equal but rather have a specific relationship, such as being greater than or less than each other. Understanding how to manipulate and solve these inequalities is crucial for problem-solving in various fields, including science, engineering, and economics. The process of solving for x involves isolating the variable on one side of the inequality, using algebraic operations while maintaining the integrity of the relationship. This means that whatever operation is performed on one side must also be performed on the other side, with careful attention paid to the direction of the inequality sign. As we embark on this journey, we'll encounter key concepts such as combining like terms, adding or subtracting constants, and multiplying or dividing by coefficients. Each step is meticulously explained to ensure clarity and comprehension. Let's begin by simplifying the given inequality and paving the way for the solution.

Step-by-Step Solution

1. Simplify the Inequality

The first step in solving for x involves simplifying the inequality. We begin with the expression: $3x + 3 - x + (-7) > 6$. Our goal is to combine like terms, which are terms that contain the same variable or are constants. In this case, we have two terms with x ($3x$ and $-x$) and two constant terms ($3$ and $-7$). By combining these terms, we can reduce the complexity of the inequality and make it easier to solve. Let's start by combining the x terms: $3x - x = 2x$. Next, we combine the constant terms: $3 + (-7) = -4$. Now, we can rewrite the inequality as: $2x - 4 > 6$. This simplified form is much easier to work with. The process of simplification is crucial in algebra because it allows us to isolate the variable of interest. By combining like terms, we reduce the number of operations and make the inequality more transparent. This step sets the stage for the subsequent operations that will ultimately lead to the solution. Remember, the key is to maintain the integrity of the inequality throughout the simplification process. Whatever operation we perform on one side must also be performed on the other side. This ensures that the relationship between the two sides remains valid. With the inequality now in a simplified form, we can proceed to the next step, which involves isolating the x term.

2. Isolate the x Term

To further isolate x, we need to eliminate the constant term on the left side of the inequality. Currently, we have $2x - 4 > 6$. To remove the $-4$, we perform the inverse operation, which is adding $4$ to both sides of the inequality. This gives us: $2x - 4 + 4 > 6 + 4$. Simplifying this, we get: $2x > 10$. Now, we have successfully isolated the term with x on one side of the inequality. This step is crucial because it brings us closer to the ultimate goal of solving for x. By adding the same value to both sides, we maintain the balance of the inequality and ensure that the relationship between the two sides remains valid. The principle of adding the same value to both sides is a fundamental concept in algebra. It allows us to manipulate equations and inequalities without altering their solutions. This principle is based on the idea that if two quantities are related by an inequality, adding the same amount to both quantities will not change the direction of the inequality. With the x term now isolated, we can proceed to the final step, which involves dividing by the coefficient of x to solve for the variable itself. This step will reveal the range of values that x can take to satisfy the given inequality.

3. Solve for x

Finally, to solve for x, we need to eliminate the coefficient of x, which is $2$. We have the inequality $2x > 10$. To isolate x, we divide both sides of the inequality by $2$. This gives us: $\frac{2x}{2} > \frac{10}{2}$. Simplifying this, we get: $x > 5$. Therefore, the solution to the inequality is $x > 5$. This means that any value of x greater than $5$ will satisfy the original inequality. The act of dividing both sides by the coefficient of x is the final step in isolating the variable and determining its value or range of values. In this case, dividing by $2$ effectively undoes the multiplication by $2$ that was previously applied to x. This step is based on the principle that if we divide both sides of an inequality by a positive number, the direction of the inequality remains the same. However, if we were to divide by a negative number, we would need to reverse the direction of the inequality. With the solution now in hand, we can confidently state that x must be greater than $5$ for the inequality to hold true. This solution can be represented graphically on a number line, where an open circle at $5$ indicates that $5$ is not included in the solution, and an arrow extending to the right indicates that all values greater than $5$ are part of the solution. In summary, by simplifying the inequality, isolating the x term, and dividing by the coefficient, we have successfully solved for x and revealed the solution set.

The Answer

The solution to the inequality $3x + 3 - x + (-7) > 6$ is $x > 5$. Therefore, the correct answer is B. $x > 5$. This solution signifies that any value of x greater than 5 will satisfy the given inequality. We arrived at this answer by meticulously simplifying the inequality, isolating the x term, and finally, dividing by its coefficient. Each step was performed with careful attention to the principles of algebra, ensuring that the integrity of the inequality was maintained throughout the process. The solution $x > 5$ can be visualized on a number line, where an open circle at 5 indicates that 5 is not included in the solution set, and an arrow extending to the right signifies that all values greater than 5 are part of the solution. This graphical representation provides a clear and intuitive understanding of the range of values that x can take. The process of solving for x in this inequality exemplifies the power of algebraic techniques in unraveling mathematical relationships. By mastering these techniques, we gain the ability to tackle a wide range of problems in mathematics and other fields. The solution $x > 5$ is not just a numerical answer; it is a testament to the elegance and precision of algebra. It represents a set of values that, when substituted into the original inequality, will always result in a true statement. This is the essence of solving inequalities: to find the values that make the relationship hold true. In conclusion, the journey of solving for x has not only provided us with the answer but also reinforced our understanding of algebraic principles and problem-solving strategies.

Key Concepts Revisited

Combining Like Terms

In the initial simplification of the inequality, we combined like terms. This involved grouping together terms with the same variable (x) and constant terms. Combining like terms is a fundamental technique in algebra that simplifies expressions and makes them easier to work with. It is based on the distributive property of multiplication over addition and subtraction. For instance, in the expression $3x - x$, we can think of this as $(3 - 1)x$, which simplifies to $2x$. Similarly, for the constant terms, $3 + (-7)$ simplifies to $-4$. The act of combining like terms reduces the complexity of the expression, making it more manageable and transparent. It allows us to focus on the essential components of the inequality and proceed with the subsequent steps in the solution process. Mastering the art of combining like terms is crucial for success in algebra, as it is a recurring theme in various mathematical problems. It is a foundational skill that lays the groundwork for more advanced algebraic techniques. In essence, combining like terms is a process of organizing and simplifying expressions, paving the way for further analysis and manipulation.

Inverse Operations

Throughout the solution process, we employed inverse operations to isolate x. Inverse operations are pairs of operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. In our case, to eliminate the $-4$ term, we added $4$ to both sides of the inequality. Similarly, to eliminate the coefficient of x, which was $2$, we divided both sides by $2$. The use of inverse operations is a cornerstone of algebraic manipulation. It allows us to selectively isolate variables and solve for their values. The principle behind this technique is that performing the same operation on both sides of an equation or inequality maintains the balance of the relationship. By strategically applying inverse operations, we can systematically unravel complex expressions and arrive at the desired solution. Understanding inverse operations is essential for anyone seeking to master algebra. It is a powerful tool that empowers us to manipulate equations and inequalities with precision and confidence. In essence, inverse operations are the key to unlocking the secrets of algebraic expressions.

Maintaining Inequality Direction

When solving inequalities, it's crucial to pay attention to the direction of the inequality sign. In most cases, the direction remains the same when performing operations on both sides. However, there is one key exception: when multiplying or dividing both sides by a negative number, the direction of the inequality must be reversed. In our example, we only divided by a positive number ($2$), so the direction of the inequality remained unchanged. However, if we had encountered a situation where we needed to divide by a negative number, we would have had to flip the inequality sign. For instance, if we had $-2x > 10$, dividing both sides by $-2$ would give us $x < -5$. The importance of maintaining the correct inequality direction cannot be overstated. Failing to do so will result in an incorrect solution. The rule of reversing the inequality sign when multiplying or dividing by a negative number is a fundamental principle in algebra that must be carefully observed. Understanding this rule ensures that we accurately represent the relationship between the variables and constants in the inequality. In summary, maintaining the inequality direction is a critical aspect of solving inequalities, and it requires careful attention to the operations being performed and the signs of the numbers involved.

Practice Problems

To solidify your understanding of solving for x in inequalities, try these practice problems:

1. Solve for x: $5x - 7 < 8$
2. Solve for x: $-2x + 3 > 11$
3. Solve for x: $4x + 9 \geq 1$

By working through these problems, you'll reinforce the concepts and techniques discussed in this article and further develop your problem-solving skills. Remember to follow the steps we've outlined: simplify the inequality, isolate the x term, and solve for x. Pay close attention to the direction of the inequality sign and be mindful of the rules for multiplying or dividing by negative numbers. Practice is the key to mastery in mathematics, and these problems provide an excellent opportunity to hone your skills. As you work through them, you'll gain confidence in your ability to solve inequalities and tackle more complex algebraic challenges. The solutions to these problems are readily available online or in textbooks, allowing you to check your work and identify areas where you may need further practice. So, dive in, challenge yourself, and enjoy the process of solving for x!

Conclusion

In conclusion, solving for x in inequalities is a fundamental skill in mathematics. By understanding the principles of simplifying expressions, using inverse operations, and maintaining the correct inequality direction, you can confidently tackle a wide range of problems. Remember, the key to success is practice and a thorough understanding of the underlying concepts. This article has provided a comprehensive guide to solving for x in the inequality $3x + 3 - x + (-7) > 6$. We've broken down the process step by step, highlighting the key techniques and principles involved. By mastering these techniques, you'll be well-equipped to solve other inequalities and tackle more advanced algebraic problems. The ability to solve for x is not just a mathematical skill; it's a problem-solving tool that can be applied in various fields. From science and engineering to economics and finance, the ability to manipulate and solve equations and inequalities is essential for critical thinking and decision-making. So, embrace the challenge, practice diligently, and enjoy the journey of mastering algebra. The rewards of mathematical proficiency are numerous, and the ability to solve for x is a crucial step along the path to success. With continued effort and dedication, you can unlock the power of algebra and apply it to solve real-world problems and achieve your goals.