Solve -7/(2x) - 8 = -5/(4x) A Step-by-Step Guide

Jun 16, 2025 by ADMIN 49 views

$Solve: $-\frac{7}{2x} - 8 = -\frac{5}{4x}$$

Introduction

In this article, we will delve into the step-by-step process of solving the equation $-\frac{7}{2x} - 8 = -\frac{5}{4x}$ . This equation falls under the category of rational equations, which are equations that contain fractions with variables in the denominator. Solving such equations requires careful manipulation to eliminate the denominators and isolate the variable. We will explore the techniques involved, including finding a common denominator, combining like terms, and ultimately solving for $x$ . This detailed explanation aims to provide a comprehensive understanding of the process, making it accessible for anyone looking to enhance their algebraic skills. Understanding how to solve rational equations is crucial not only in mathematics but also in various fields of science and engineering, where such equations often arise in modeling real-world phenomena. This guide will break down each step, ensuring clarity and accuracy in the solution.

Detailed Solution

To solve the equation $-\frac{7}{2x} - 8 = -\frac{5}{4x}$ , the first step is to eliminate the fractions. This can be achieved by finding the least common denominator (LCD) of the fractions in the equation. The denominators in our equation are $2x$ and $4x$ . To find the LCD, we look for the smallest expression that is divisible by both $2x$ and $4x$ . In this case, the LCD is $4x$ . Multiplying both sides of the equation by the LCD will clear the fractions.

Identify the LCD: The denominators are $2x$ and $4x$ . The LCD is $4x$ .
Multiply both sides by the LCD:

$4x(-\frac{7}{2x} - 8) = 4x(-\frac{5}{4x})$
Distribute $4x$ on the left side:

$4x(-\frac{7}{2x}) - 4x(8) = 4x(-\frac{5}{4x})$
Simplify each term:

$-14 - 32x = -5$
Isolate the term with $x$ :

Add 14 to both sides:

$-32x = -5 + 14$

$-32x = 9$
Solve for $x$ :

Divide both sides by -32:

$x = \frac{9}{-32}$

$x = -\frac{9}{32}$

Therefore, the solution to the equation $-\frac{7}{2x} - 8 = -\frac{5}{4x}$ is $x = -\frac{9}{32}$ .

Step-by-Step Breakdown

Step 1: Clearing the Fractions

The crucial first step in solving the equation $-\frac{7}{2x} - 8 = -\frac{5}{4x}$ is to eliminate the fractions. Fractions can make equations more complex and harder to manipulate, so clearing them simplifies the process significantly. To do this, we identify the least common denominator (LCD) of the denominators present in the equation. In this case, the denominators are $2x$ and $4x$ . The LCD is the smallest multiple that both denominators divide into evenly. To find it, we consider the multiples of each denominator. The multiples of $2x$ are $2x$ , $4x$ , $6x$ , and so on, while the multiples of $4x$ are $4x$ , $8x$ , $12x$ , and so on. The smallest multiple that appears in both lists is $4x$ , so the LCD is $4x$ . Once we have the LCD, we multiply both sides of the equation by $4x$ . This ensures that we maintain the equality while eliminating the fractions. Multiplying each term by $4x$ will allow us to cancel out the denominators.

The equation becomes:

$4x(-\frac{7}{2x} - 8) = 4x(-\frac{5}{4x})$

Step 2: Distributing and Simplifying

After multiplying both sides of the equation by the least common denominator (LCD), which is $4x$ , the next step is to distribute $4x$ on the left side of the equation. Distributing involves multiplying $4x$ by each term inside the parentheses. This gives us:

$4x(-\frac{7}{2x}) - 4x(8)$

When we multiply $4x$ by $-\frac{7}{2x}$ , the $x$ terms cancel out, and we simplify the fraction. $4x$ divided by $2x$ is 2, so we have $2 * -7$ , which equals -14. Next, we multiply $4x$ by 8, resulting in $-32x$ . The left side of the equation now simplifies to $-14 - 32x$ . On the right side of the equation, we have $4x(-\frac{5}{4x})$ . Here, the $4x$ in the numerator and the denominator cancel each other out, leaving us with -5. So, the right side of the equation simplifies to -5. Putting it all together, the equation after distributing and simplifying becomes:

$-14 - 32x = -5$

Step 3: Isolating the Variable Term

Once the equation is simplified to $-14 - 32x = -5$ , the next key step is to isolate the term containing the variable, which in this case is $-32x$ . Isolating a variable term means getting it by itself on one side of the equation. To achieve this, we need to eliminate any constants or other terms that are on the same side as the variable term. In our equation, the constant term on the left side is -14. To eliminate -14, we perform the inverse operation, which is adding 14 to both sides of the equation. This maintains the balance of the equation while moving the constant term to the right side. Adding 14 to both sides of the equation $-14 - 32x = -5$ gives us:

$-14 + 14 - 32x = -5 + 14$

The -14 and +14 on the left side cancel each other out, leaving us with just $-32x$ . On the right side, -5 plus 14 equals 9. So, the equation simplifies to:

$-32x = 9$

This step is crucial because it brings us closer to solving for $x$ by separating the variable term from the constants.

Step 4: Solving for $x$

After isolating the variable term, the final step in solving the equation $-32x = 9$ is to solve for $x$ . To do this, we need to get $x$ completely by itself on one side of the equation. Currently, $x$ is being multiplied by -32. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by -32. This will isolate $x$ on the left side and give us the value of $x$ on the right side. Dividing both sides of $-32x = 9$ by -32 gives us:

$\frac{-32x}{-32} = \frac{9}{-32}$

On the left side, the -32 in the numerator and the denominator cancel each other out, leaving us with $x$ . On the right side, we have the fraction $\frac{9}{-32}$ . This fraction can be simplified by expressing it as a negative fraction, which is $-\frac{9}{32}$ . So, the solution for $x$ is:

$x = -\frac{9}{32}$

This final step completes the solution process, giving us the value of $x$ that satisfies the original equation.

Verification

To ensure the solution $x = -\frac{9}{32}$ is correct, it is important to substitute this value back into the original equation and check if both sides of the equation are equal. The original equation is:

$-\frac{7}{2x} - 8 = -\frac{5}{4x}$

Substitute $x = -\frac{9}{32}$ into the equation:

$-\frac{7}{2(-\frac{9}{32})} - 8 = -\frac{5}{4(-\frac{9}{32})}$

First, simplify the denominators:

$2(-\frac{9}{32}) = -\frac{18}{32} = -\frac{9}{16}$

$4(-\frac{9}{32}) = -\frac{36}{32} = -\frac{9}{8}$

Now, substitute these simplified denominators back into the equation:

$-\frac{7}{-\frac{9}{16}} - 8 = -\frac{5}{-\frac{9}{8}}$

Dividing by a fraction is the same as multiplying by its reciprocal, so we rewrite the equation as:

$-7(-\frac{16}{9}) - 8 = -5(-\frac{8}{9})$

Multiply:

$\frac{112}{9} - 8 = \frac{40}{9}$

Convert 8 to a fraction with a denominator of 9:

$8 = \frac{72}{9}$

Substitute this back into the equation:

$\frac{112}{9} - \frac{72}{9} = \frac{40}{9}$

Subtract:

$\frac{40}{9} = \frac{40}{9}$

Since both sides of the equation are equal, the solution $x = -\frac{9}{32}$ is verified to be correct.

Common Mistakes to Avoid

When solving rational equations like $-\frac{7}{2x} - 8 = -\frac{5}{4x}$ , it’s essential to be aware of common mistakes that can lead to incorrect solutions. One frequent error is failing to distribute the least common denominator (LCD) correctly. Remember, when you multiply both sides of the equation by the LCD, each term must be multiplied. Forgetting to multiply every term can disrupt the balance of the equation and result in an incorrect answer. Another common mistake is incorrectly simplifying fractions or making errors in arithmetic operations, especially when dealing with negative signs. Double-checking each step of your calculations can help prevent these errors. Additionally, it’s crucial to verify the solution by substituting it back into the original equation. This step helps catch any mistakes made during the solving process. Overlooking this verification step can lead to accepting an extraneous solution, which is a value that satisfies the transformed equation but not the original. By being mindful of these common pitfalls—incorrectly distributing the LCD, making arithmetic errors, and neglecting to verify the solution—you can improve your accuracy and confidence in solving rational equations.

Conclusion

In this article, we have thoroughly explored the process of solving the equation $-\frac{7}{2x} - 8 = -\frac{5}{4x}$ . We began by identifying the need to eliminate fractions, which led us to find the least common denominator (LCD). Multiplying both sides of the equation by the LCD, $4x$ , allowed us to clear the fractions and simplify the equation. We then distributed and combined like terms, isolating the variable term to one side. By dividing both sides by the coefficient of $x$ , we successfully solved for $x$ , obtaining the solution $x = -\frac{9}{32}$ . To ensure the accuracy of our solution, we verified it by substituting it back into the original equation, confirming that both sides were equal. We also highlighted common mistakes to avoid, such as incorrect distribution, arithmetic errors, and neglecting to verify the solution. Understanding these steps and avoiding common pitfalls will equip you with the skills to confidently solve similar rational equations. Mastering the techniques discussed here is not only valuable for academic pursuits but also for various real-world applications where mathematical problem-solving is essential. This comprehensive guide aims to provide a clear and reliable approach to solving rational equations, empowering you to tackle these problems with precision and understanding.