Solving For X In The Equation 1.5(x + 4) - 3 = 4.5(x - 2)

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Introduction

In this article, we will delve into the process of finding the value of x in the equation 1.5(x + 4) - 3 = 4.5(x - 2). This is a fundamental algebraic problem that requires a step-by-step approach to solve accurately. Understanding how to solve such equations is crucial for various mathematical and real-world applications. Whether you're a student learning algebra or someone looking to refresh your math skills, this guide will provide a clear and concise method to solve this type of equation. We will break down each step, ensuring that you understand the logic and reasoning behind the solution. By the end of this article, you will not only know the value of x but also grasp the underlying principles of algebraic problem-solving.

Algebraic equations like this one are the backbone of many mathematical concepts. They appear in various fields, from physics and engineering to economics and computer science. The ability to manipulate and solve these equations is a valuable skill that enhances problem-solving capabilities across different disciplines. This article serves as a comprehensive guide to not just solving the specific equation at hand, but also to understanding the general methodology that can be applied to similar problems. By following the steps outlined here, you can build a strong foundation in algebra and improve your overall mathematical proficiency. So, let's dive into the solution and unravel the mystery of x in this equation.

Understanding the importance of each step is key to mastering algebraic equations. Often, students make mistakes due to overlooking a critical operation or misapplying a rule. Therefore, we will emphasize the rationale behind each step, ensuring that you not only arrive at the correct answer but also understand why that answer is correct. This approach helps to build a deeper understanding of algebra and enhances your ability to tackle more complex problems in the future. Furthermore, we will highlight common pitfalls and mistakes that students often make, providing you with the knowledge to avoid them. By the end of this article, you will have a thorough understanding of how to solve this equation and a solid foundation for future algebraic endeavors. This skill is not just about getting the right answer; it's about developing a way of thinking that will benefit you in numerous areas of life.

Step-by-Step Solution

To solve the equation 1.5(x + 4) - 3 = 4.5(x - 2), we will follow a series of algebraic steps. The primary goal is to isolate the variable x on one side of the equation. This involves simplifying the equation by distributing, combining like terms, and using inverse operations to move terms around. Each step is crucial to maintaining the balance of the equation and ensuring an accurate solution. Let's break down the process into manageable steps:

1. Distribute the Constants

The first step in solving this equation is to distribute the constants outside the parentheses. This means multiplying 1.5 by both x and 4 on the left side, and multiplying 4.5 by both x and -2 on the right side. This is a fundamental application of the distributive property, which states that a(b + c) = ab + ac. Applying this property correctly is crucial for simplifying the equation and moving closer to isolating x. Let's perform the multiplication:

  1. 5(x + 4) becomes 1.5x + 1.5 * 4, which simplifies to 1.5x + 6.
  2. 5(x - 2) becomes 4.5x - 4.5 * 2, which simplifies to 4.5x - 9.

After distributing, the equation now looks like this: 1.5x + 6 - 3 = 4.5x - 9. This is a significant step forward, as we have eliminated the parentheses and made the equation more manageable. The next step involves combining like terms to further simplify the equation. Understanding the distributive property is essential not only for this problem but for various algebraic manipulations. By mastering this step, you are laying a solid foundation for solving more complex equations.

2. Combine Like Terms

After distributing the constants, the next logical step is to combine like terms. This involves identifying terms on the same side of the equation that can be added or subtracted. In our equation, 1.5x + 6 - 3 = 4.5x - 9, we can combine the constant terms 6 and -3 on the left side. This simplification helps to make the equation cleaner and easier to work with. By combining like terms, we reduce the number of individual terms in the equation, which streamlines the process of isolating x. Let's perform the combination:

  • On the left side, 6 - 3 simplifies to 3.

So, the equation becomes: 1.5x + 3 = 4.5x - 9. Now, we have a much simpler equation to work with. Combining like terms is a fundamental skill in algebra, and it's essential for efficiently solving equations. This step reduces the complexity of the equation and sets the stage for the next phase of isolating the variable x. By mastering this technique, you can significantly improve your ability to solve algebraic problems. The next step will involve moving variable terms to one side and constant terms to the other.

3. Move Variable Terms to One Side

To further isolate x, we need to move all the variable terms to one side of the equation. In our case, the equation is 1.5x + 3 = 4.5x - 9. It's generally easier to move the smaller variable term to the side with the larger variable term to avoid dealing with negative coefficients. In this equation, 1.5x is smaller than 4.5x, so we will subtract 1.5x from both sides of the equation. This maintains the balance of the equation while moving the variable terms closer to isolation. Let's perform the subtraction:

  • Subtracting 1.5x from both sides gives us: 1.5x + 3 - 1.5x = 4.5x - 9 - 1.5x.
  • Simplifying this, we get: 3 = 3x - 9.

Now, all the variable terms are on the right side of the equation, and we have a simpler equation to solve. Moving variable terms to one side is a crucial step in solving linear equations. It allows us to group the terms containing x together, which is essential for isolating x. This step demonstrates the importance of using inverse operations to manipulate equations while maintaining equality. The next step will involve moving the constant terms to the other side of the equation.

4. Move Constant Terms to the Other Side

Now that we have the variable terms on one side, the next step is to move the constant terms to the other side. Our equation is currently 3 = 3x - 9. To isolate the term with x, we need to eliminate the -9 on the right side. We can do this by adding 9 to both sides of the equation. This maintains the balance of the equation and brings us closer to isolating x. Adding the constant term is another application of using inverse operations to solve equations. Let's perform the addition:

  • Adding 9 to both sides gives us: 3 + 9 = 3x - 9 + 9.
  • Simplifying this, we get: 12 = 3x.

Now, we have a very simple equation where the variable term is almost isolated. Moving constant terms to one side is a key step in solving linear equations, as it groups all the constant terms together, making it easier to isolate the variable. This step further demonstrates the importance of maintaining balance in an equation by performing the same operation on both sides. The final step will be to divide both sides by the coefficient of x to solve for x.

5. Divide to Isolate x

Finally, to isolate x, we need to divide both sides of the equation by the coefficient of x. Our equation is currently 12 = 3x. The coefficient of x is 3, so we will divide both sides by 3. This is the final step in solving for x, and it will give us the value of x that satisfies the original equation. Dividing by the coefficient is the inverse operation of multiplication, and it's the key to isolating the variable. Let's perform the division:

  • Dividing both sides by 3 gives us: 12 / 3 = 3x / 3.
  • Simplifying this, we get: 4 = x.

Therefore, the value of x that satisfies the equation is 4. This final step completes the process of solving the equation, and it highlights the importance of using inverse operations to isolate the variable. By mastering this step, you can confidently solve a wide range of linear equations. The solution x = 4 means that if we substitute 4 for x in the original equation, both sides of the equation will be equal.

Verification

To ensure our solution is correct, it's essential to verify the value of x by substituting it back into the original equation. This step is crucial for catching any potential errors made during the solving process. Substituting the solution back into the original equation allows us to check if both sides of the equation are equal, confirming the accuracy of our solution. Let's substitute x = 4 into the original equation:

Original equation: 1. 5(x + 4) - 3 = 4.5(x - 2)

Substitute x = 4:

  1. 5(4 + 4) - 3 = 4.5(4 - 2)

Now, let's simplify both sides of the equation:

Left side:

  1. 5(8) - 3
  2. 2 - 3
  3. 0

Right side:

  1. 5(2)
  2. 0

Since both sides of the equation are equal (9 = 9), our solution x = 4 is correct. Verification is a critical step in problem-solving, as it provides assurance that the solution is accurate. By substituting the solution back into the original equation, we can confirm that the value of x satisfies the equation. This step reinforces the importance of accuracy and attention to detail in algebraic problem-solving. If the two sides of the equation were not equal after substitution, it would indicate an error in the solving process, prompting us to review our steps.

Conclusion

In conclusion, we have successfully solved the equation 1.5(x + 4) - 3 = 4.5(x - 2) and found that the value of x is 4. We achieved this by following a step-by-step approach that included distributing constants, combining like terms, moving variable terms to one side, moving constant terms to the other side, and finally, dividing to isolate x. Each step was crucial in simplifying the equation and arriving at the correct solution. Furthermore, we verified our solution by substituting x = 4 back into the original equation, confirming its accuracy.

This exercise demonstrates the fundamental principles of algebraic problem-solving and highlights the importance of a systematic approach. By breaking down the problem into manageable steps, we were able to tackle it effectively and efficiently. Understanding these principles is essential for solving more complex algebraic problems in the future. The ability to manipulate equations and isolate variables is a valuable skill that extends beyond mathematics and applies to various fields.

By mastering the techniques discussed in this article, you can build a strong foundation in algebra and enhance your problem-solving abilities. Remember, practice is key to improving your skills, so continue to work on similar problems to solidify your understanding. Algebraic equations are a fundamental part of mathematics, and the ability to solve them is a valuable asset in both academic and real-world scenarios. This article has provided you with a clear and concise method for solving this type of equation, and with practice, you can confidently tackle any algebraic challenge.