Solving (0.5x - 4) / (2.4x + 6) = -5 / 3 A Step-by-Step Guide

In this article, we will delve into the step-by-step process of solving the equation (0.5x - 4) / (2.4x + 6) = -5 / 3. This type of equation, involving fractions and variables, often appears in algebra and requires a systematic approach to find the correct solution. Understanding how to solve such equations is crucial for mastering algebraic concepts and succeeding in mathematics. We will break down each step, providing explanations and insights to ensure a clear understanding. This comprehensive guide will not only help you solve this specific equation but also equip you with the skills to tackle similar problems confidently. Let’s embark on this mathematical journey and unravel the solution together, ensuring that every concept is crystal clear and easily applicable in future scenarios.

1. Introduction to Solving Algebraic Equations

Understanding the Basics of Algebraic Equations

Before diving into the specifics of solving our equation, it’s essential to grasp the fundamental principles of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions often contain variables, which are symbols (usually letters like x, y, or z) representing unknown values. Solving an algebraic equation means finding the value(s) of the variable(s) that make the equation true. This involves manipulating the equation using algebraic rules and operations to isolate the variable on one side. Understanding the basic properties of equality, such as the addition, subtraction, multiplication, and division properties, is crucial for this process. These properties allow us to perform the same operation on both sides of the equation without changing its balance. For instance, if we add the same number to both sides of an equation, the equality remains valid. Similarly, multiplying both sides by the same non-zero number preserves the equality. Mastering these basics is the cornerstone of solving more complex equations, including the one we are about to tackle.

The Importance of Order of Operations

When solving equations, adhering to the correct order of operations is paramount. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) serves as a handy guide for this. It dictates the sequence in which operations should be performed to ensure accuracy. First, we address any expressions within parentheses. Next, we handle exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). Neglecting the order of operations can lead to incorrect simplifications and, consequently, an inaccurate solution. In our equation, we'll need to carefully consider how each operation affects the terms and maintain the correct order to arrive at the right answer. Understanding and applying PEMDAS is not just about following a rule; it's about ensuring logical consistency in our mathematical steps. This disciplined approach helps prevent errors and makes the process of solving equations more streamlined and understandable.

2. Step-by-Step Solution of (0.5x - 4) / (2.4x + 6) = -5 / 3

Step 1: Cross-Multiplication

The first step in solving the equation (0.5x - 4) / (2.4x + 6) = -5 / 3 is to eliminate the fractions. We achieve this by cross-multiplication, a technique that involves multiplying the numerator of one fraction by the denominator of the other and vice versa. This process transforms the equation into a more manageable form without fractions. In our case, we multiply (0.5x - 4) by 3 and (2.4x + 6) by -5. This yields the equation 3 * (0.5x - 4) = -5 * (2.4x + 6). Cross-multiplication is a powerful tool for dealing with equations involving fractions, as it simplifies the equation while preserving the equality. It’s crucial to perform this step accurately to ensure the subsequent steps are based on a correct foundation. By eliminating fractions early on, we make the equation easier to manipulate and solve. This method is widely used in algebra and is essential for solving various types of equations.

Step 2: Expanding Both Sides

Following cross-multiplication, our equation stands as 3 * (0.5x - 4) = -5 * (2.4x + 6). The next step involves expanding both sides of the equation by applying the distributive property. This property states that a(b + c) = ab + ac, meaning we multiply the term outside the parentheses by each term inside. On the left side, we multiply 3 by both 0.5x and -4, resulting in 1.5x - 12. On the right side, we multiply -5 by both 2.4x and 6, giving us -12x - 30. Thus, the expanded equation becomes 1.5x - 12 = -12x - 30. Expanding the equation is crucial for removing the parentheses and combining like terms in the subsequent steps. Accuracy in applying the distributive property is vital; any error here can propagate through the rest of the solution. This step is a fundamental technique in algebra and is frequently used to simplify equations.

Step 3: Collecting Like Terms

After expanding the equation to 1.5x - 12 = -12x - 30, our next goal is to collect like terms. This involves gathering all the terms containing the variable x on one side of the equation and all the constant terms on the other side. To do this, we can add 12x to both sides of the equation to move the x term from the right side to the left, and we can add 12 to both sides to move the constant term from the left side to the right. Adding 12x to both sides gives us 1.5x + 12x - 12 = -30, and adding 12 to both sides then gives us 1.5x + 12x = -30 + 12. Simplifying these operations, we get 13.5x = -18. Collecting like terms is a critical step in isolating the variable and simplifying the equation. This process makes it easier to see the relationship between the variable and the constants, paving the way for the final steps of solving for x. This technique is a cornerstone of algebraic manipulation and is essential for solving a wide range of equations.

Step 4: Isolating the Variable

Now that we have the equation simplified to 13.5x = -18, the next crucial step is to isolate the variable x. To do this, we need to undo the operation that is currently being applied to x, which in this case is multiplication by 13.5. The inverse operation of multiplication is division, so we divide both sides of the equation by 13.5. This gives us x = -18 / 13.5. Performing this division will give us the value of x. Isolating the variable is the ultimate goal of solving any algebraic equation. It involves using inverse operations to get the variable by itself on one side of the equation, allowing us to determine its value. This step demonstrates a fundamental principle of algebra: performing the same operation on both sides of an equation maintains its balance and helps us solve for the unknown.

Step 5: Simplifying the Solution

After isolating the variable, we arrive at the solution x = -18 / 13.5. The final step is to simplify this fraction to its simplest form. To do this, we can divide both the numerator and the denominator by their greatest common divisor. First, let's convert the decimal in the denominator to a whole number by multiplying both the numerator and the denominator by 10, which gives us x = -180 / 135. Now, we can see that both 180 and 135 are divisible by 45. Dividing both by 45, we get x = -4 / 3. This is the simplified form of our solution. Simplifying the solution ensures that we present the answer in its most concise and understandable form. It also makes it easier to work with the solution in further calculations or applications. Simplifying fractions is an essential skill in algebra and helps to maintain clarity and precision in mathematical expressions.

3. Verification of the Solution

Plugging the Solution Back into the Original Equation

After obtaining the solution x = -4 / 3, it's crucial to verify its correctness. This is done by substituting the value of x back into the original equation, (0.5x - 4) / (2.4x + 6) = -5 / 3, and checking if both sides of the equation are equal. This process ensures that the solution we found satisfies the original equation and that no errors were made during the solving process. Substituting x = -4 / 3 into the equation, we get (0.5 * (-4 / 3) - 4) / (2.4 * (-4 / 3) + 6). Now, we simplify the expression to check if it equals -5 / 3. Verifying the solution is a critical step in problem-solving. It provides confidence in the correctness of the answer and helps identify any potential mistakes. This practice reinforces the understanding of the equation and the solution process.

Step-by-Step Verification Process

Let’s perform the step-by-step verification of our solution x = -4 / 3 in the original equation (0.5x - 4) / (2.4x + 6) = -5 / 3. First, we substitute x = -4 / 3 into the equation: (0. 5 * (-4 / 3) - 4) / (2.4 * (-4 / 3) + 6). Next, we simplify the numerator and the denominator separately. In the numerator, 0.5 * (-4 / 3) equals -2 / 3, so the numerator becomes -2 / 3 - 4. Converting 4 to a fraction with a denominator of 3, we get -2 / 3 - 12 / 3, which simplifies to -14 / 3. In the denominator, 2.4 * (-4 / 3) equals -3.2, so the denominator becomes -3.2 + 6. This simplifies to 2.8. Now, our equation looks like (-14 / 3) / 2.8 = -5 / 3. To divide by 2.8, we can multiply by its reciprocal, which is 1 / 2.8. So, we have (-14 / 3) * (1 / 2.8). To make the calculation easier, we can convert 2.8 to a fraction, which is 28 / 10 or 14 / 5. Thus, we have (-14 / 3) * (5 / 14). Simplifying this, we cancel out the 14s and get -5 / 3, which matches the right side of the original equation. This step-by-step verification confirms that our solution x = -4 / 3 is indeed correct. This detailed process highlights the importance of careful calculation and simplification in verifying solutions.

4. Common Mistakes and How to Avoid Them

Identifying Common Errors in Solving Equations

Solving algebraic equations can be challenging, and certain errors frequently occur. One common mistake is incorrect application of the distributive property, particularly when dealing with negative signs. For example, failing to distribute the negative sign across all terms within the parentheses can lead to an incorrect expansion of the equation. Another common error is making mistakes when combining like terms, such as adding or subtracting terms incorrectly. Additionally, students often make errors when performing cross-multiplication, especially when dealing with more complex fractions or expressions. It's also common to forget the order of operations (PEMDAS), leading to incorrect simplification. Another frequent mistake is failing to verify the solution by plugging it back into the original equation, which can catch calculation errors. Identifying these common errors is the first step in preventing them. By being aware of these pitfalls, students can approach equation-solving with greater caution and precision.

Strategies for Avoiding Mistakes

To avoid common mistakes when solving equations, several strategies can be employed. First, always double-check the application of the distributive property, paying close attention to negative signs. Write out each step explicitly to ensure accuracy in expansion. When combining like terms, be methodical and group similar terms together before performing the operations. This helps prevent errors in addition and subtraction. When cross-multiplying, ensure that the multiplication is performed correctly and that no terms are missed. Adhering strictly to the order of operations (PEMDAS) is crucial; use it as a checklist throughout the solving process. Finally, always verify the solution by substituting it back into the original equation. This step is invaluable for catching errors and building confidence in the answer. These strategies promote accuracy and clarity in the equation-solving process. By consistently applying these techniques, students can minimize errors and improve their algebraic skills.

5. Conclusion

Recap of the Solution Process

In this comprehensive guide, we have meticulously walked through the process of solving the equation (0.5x - 4) / (2.4x + 6) = -5 / 3. We began by understanding the basics of algebraic equations and the importance of the order of operations. Then, we tackled the equation step by step, starting with cross-multiplication to eliminate fractions, followed by expanding both sides using the distributive property. We then collected like terms, bringing all x terms to one side and constants to the other. Next, we isolated the variable x by dividing both sides by its coefficient, and finally, we simplified the solution to its lowest terms. To ensure the correctness of our solution, we verified it by plugging it back into the original equation and confirming that both sides matched. This systematic approach not only solved the equation but also provided a clear understanding of the underlying algebraic principles.

Importance of Mastering Algebraic Equations

Mastering the skills to solve algebraic equations is of paramount importance in mathematics and beyond. Algebraic equations form the foundation for more advanced mathematical concepts, such as calculus, linear algebra, and differential equations. They are also essential in various fields, including physics, engineering, economics, and computer science. The ability to solve equations accurately and efficiently is crucial for problem-solving in these disciplines. Furthermore, the logical thinking and analytical skills developed through solving algebraic equations are transferable to many other areas of life. Whether you are balancing a budget, making investment decisions, or analyzing data, the problem-solving skills honed by algebra will prove invaluable. Mastering algebraic equations not only opens doors to advanced mathematical studies but also equips you with essential skills for success in a wide range of endeavors.