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

In mathematics, solving for a variable is a fundamental skill. It involves isolating the variable on one side of the equation to determine its value. This article will delve into the process of solving for x in the equation 4(x+2) = 3(x-2). We will break down each step, providing a clear and concise explanation to enhance your understanding of algebraic equations. Whether you are a student learning algebra for the first time or someone looking to refresh your knowledge, this guide will equip you with the necessary tools to tackle similar problems confidently. Understanding how to solve for x not only strengthens your mathematical foundation but also helps in various real-world applications where equations need to be solved to find unknown quantities. So, let's embark on this mathematical journey and demystify the process of solving for x.

Before we dive into the steps to solve the equation 4(x+2) = 3(x-2), it's essential to understand what this equation represents. An equation is a mathematical statement that asserts the equality of two expressions. In this case, the expression on the left side, 4(x+2), is equal to the expression on the right side, 3(x-2). The variable x represents an unknown value that we need to find. Solving for x means determining the value of x that makes the equation true. The structure of the equation involves parentheses, which indicate multiplication. The numbers outside the parentheses (4 and 3) are coefficients that multiply the expressions inside the parentheses. The expressions inside the parentheses involve x and constants (2 and -2). To solve this equation, we will use the principles of algebra, including the distributive property and combining like terms. Understanding the components of the equation and how they relate to each other is crucial for effectively solving it. This foundational knowledge will guide us through each step of the solution process, ensuring a clear understanding of why we perform each operation. So, let's begin our journey by unraveling the mysteries of this equation.

To solve the equation 4(x+2) = 3(x-2), we will follow a step-by-step approach, ensuring clarity and accuracy in our solution. Each step is crucial in isolating x and finding its value.

Step 1: Distribute the Coefficients

The first step is to apply the distributive property to both sides of the equation. The distributive property states that a(b+c) = ab + ac. This means we need to multiply the coefficient outside the parentheses by each term inside the parentheses.

On the left side, we have 4(x+2). Multiplying 4 by both x and 2 gives us 4x + 42, which simplifies to 4x + 8.

On the right side, we have 3(x-2). Multiplying 3 by both x and -2 gives us 3x - 32, which simplifies to 3x - 6.

So, after distributing the coefficients, our equation becomes:

4x + 8 = 3x - 6

Step 2: Rearrange the Equation

Now that we have removed the parentheses, our next goal is to rearrange the equation so that all the terms involving x are on one side and the constant terms are on the other side. This is done by adding or subtracting terms from both sides of the equation.

To move the x term from the right side to the left side, we subtract 3x from both sides:

4x + 8 - 3x = 3x - 6 - 3x

This simplifies to:

x + 8 = -6

Next, we need to move the constant term from the left side to the right side. To do this, we subtract 8 from both sides:

x + 8 - 8 = -6 - 8

This simplifies to:

x = -14

Step 3: Isolate x

After rearranging the equation, we have successfully isolated x on one side. The equation now reads:

x = -14

This means that the value of x that satisfies the original equation is -14. We have found our solution!

Step 4: Verify the Solution

To ensure our solution is correct, it's always a good practice to verify it by substituting the value of x back into the original equation.

Our original equation was:

4(x+2) = 3(x-2)

Substituting x = -14, we get:

4(-14+2) = 3(-14-2)

Simplifying the expressions inside the parentheses:

4(-12) = 3(-16)

Multiplying the coefficients:

-48 = -48

Since both sides of the equation are equal, our solution x = -14 is correct.

In this article, we have successfully solved for x in the equation 4(x+2) = 3(x-2). We followed a systematic approach, breaking down the problem into manageable steps. First, we distributed the coefficients to remove the parentheses. Then, we rearranged the equation to group like terms together. After that, we isolated x to find its value. Finally, we verified our solution by substituting it back into the original equation. The solution we found is x = -14. This process demonstrates the importance of understanding algebraic principles and applying them methodically. Solving for variables is a fundamental skill in mathematics, and mastering it opens the door to solving more complex problems. Remember, practice is key to improving your problem-solving skills. So, keep practicing, and you'll become more confident in your ability to tackle algebraic equations.

To solidify your understanding of solving linear equations, it's beneficial to practice with additional examples. Here are a few practice problems similar to the one we solved in this article:

1. 5(x - 3) = 2(x + 6)
2. 3(2x + 1) = 4(x - 2)
3. 6(x + 4) = 2(3x - 5)
4. 4(2x - 3) = 5(x + 1)
5. 2(5x + 2) = 3(3x - 4)

Try solving these equations on your own, following the steps we discussed earlier. Remember to distribute, rearrange, isolate, and verify your solutions. Working through these problems will help you build confidence and proficiency in solving for x. Additionally, you can seek out more practice problems in textbooks, online resources, or from your instructors. The more you practice, the better you will become at solving algebraic equations. So, challenge yourself, stay persistent, and enjoy the process of learning mathematics.

Solving for variables, like x in our example equation, isn't just a theoretical exercise; it has numerous real-world applications. Algebraic equations are used to model and solve problems in various fields, including science, engineering, finance, and economics. For instance, in physics, equations are used to calculate the trajectory of a projectile or the current in an electrical circuit. In finance, equations can help determine the rate of return on an investment or the amount of a loan payment. In economics, they can be used to model supply and demand curves. Even in everyday situations, we use algebraic thinking to solve problems. For example, if you're trying to figure out how much to charge for a service, you might set up an equation to calculate your costs and desired profit. Similarly, if you're planning a road trip, you might use an equation to estimate how long it will take to reach your destination based on your speed and distance. The ability to solve for variables allows us to make informed decisions and solve problems effectively in a wide range of contexts. By mastering these skills, you're not just learning math; you're developing a valuable tool for navigating the world around you. The applications are virtually limitless, making algebra a crucial skill for anyone pursuing higher education or a career in a STEM field.

In conclusion, solving for x in the equation 4(x+2) = 3(x-2) is a fundamental exercise in algebra that highlights the core principles of equation solving. We systematically applied the distributive property, rearranged terms, isolated the variable, and verified our solution. This process not only provided us with the answer, x = -14, but also reinforced the importance of each step in ensuring accuracy and understanding. Algebraic problem-solving is a critical skill that extends far beyond the classroom. It forms the backbone of many scientific, engineering, and financial calculations. The ability to manipulate equations, isolate variables, and find solutions is essential for anyone pursuing a career in these fields. Moreover, algebraic thinking enhances problem-solving skills in general, allowing individuals to approach complex issues with a logical and structured mindset. By mastering algebra, you gain a powerful tool for analyzing and solving problems in a variety of contexts. It fosters critical thinking, attention to detail, and the ability to break down complex problems into smaller, more manageable steps. As you continue your mathematical journey, remember that algebra is not just about memorizing formulas; it's about developing a way of thinking that will serve you well in all aspects of life. The problem-solving skills acquired through algebra are transferable to various situations, making it an invaluable asset in both personal and professional endeavors.