Solving Absolute Value Equation |5/(x-5)| = 15

In this article, we will delve into the process of solving the absolute value equation $\left|\frac{5}{x-5}\right|=15$. Absolute value equations require a careful approach, as the absolute value of a number represents its distance from zero, meaning there are often two possible solutions to consider. Understanding the concept of absolute value and applying algebraic techniques are crucial for finding the correct solutions. This problem involves fractions and absolute values, making it a valuable exercise in strengthening algebraic skills. We will break down the problem step-by-step, explaining the reasoning behind each operation to make the solution clear and understandable.

Understanding Absolute Value Equations

Before we jump into solving the specific equation, let's recap the basics of absolute value. The absolute value of a number, denoted by $|a|$, is its distance from zero on the number line. This means that $|a|$ is always non-negative. For example, $|3| = 3$ and $|-3| = 3$. When dealing with absolute value equations, we need to consider two cases: the expression inside the absolute value can be either positive or negative, but its absolute value remains the same. Therefore, if $|x| = a$, then either $x = a$ or $x = -a$. This principle is fundamental to solving absolute value equations effectively. In our given equation, $\left|\frac{5}{x-5}\right|=15$, we need to apply this principle by considering both positive and negative scenarios for the expression inside the absolute value bars. By carefully analyzing each case, we can identify all possible solutions for $x$ that satisfy the equation.

Step-by-Step Solution

To solve the equation $\left|\frac{5}{x-5}\right|=15$, we need to consider two cases based on the definition of absolute value:

Case 1: The expression inside the absolute value is positive.

If $\frac{5}{x-5}$ is positive, then the absolute value is simply the expression itself. So, we have:

$\frac{5}{x-5} = 15$

To solve this equation, we can first multiply both sides by $(x-5)$ to eliminate the fraction:

$5 = 15(x-5)$

Now, distribute the 15 on the right side:

$5 = 15x - 75$

Add 75 to both sides:

$80 = 15x$

Finally, divide both sides by 15:

$x = \frac{80}{15}$

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 5:

$x = \frac{16}{3}$

Case 2: The expression inside the absolute value is negative.

If $\frac{5}{x-5}$ is negative, then the absolute value is the negation of the expression. So, we have:

$-\frac{5}{x-5} = 15$

Multiply both sides by -1:

$\frac{5}{x-5} = -15$

Now, multiply both sides by $(x-5)$ to eliminate the fraction:

$5 = -15(x-5)$

Distribute the -15 on the right side:

$5 = -15x + 75$

Subtract 75 from both sides:

$-70 = -15x$

Divide both sides by -15:

$x = \frac{-70}{-15}$

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 5:

$x = \frac{14}{3}$

Therefore, the solutions to the equation are $x = \frac{16}{3}$ and $x = \frac{14}{3}$.

Verification of Solutions

To ensure our solutions are correct, we should substitute each value of $x$ back into the original equation and check if it holds true.

Verification for x = 16/3:

$\left|\frac{5}{(\frac{16}{3})-5}\right| = \left|\frac{5}{(\frac{16}{3})-(\frac{15}{3})}\right| = \left|\frac{5}{\frac{1}{3}}\right| = |5 \cdot 3| = |15| = 15$

So, $x = \frac{16}{3}$ is a valid solution.

Verification for x = 14/3:

$\left|\frac{5}{(\frac{14}{3})-5}\right| = \left|\frac{5}{(\frac{14}{3})-(\frac{15}{3})}\right| = \left|\frac{5}{-\frac{1}{3}}\right| = |5 \cdot (-3)| = |-15| = 15$

So, $x = \frac{14}{3}$ is also a valid solution.

Both solutions satisfy the original equation, confirming our calculations.

Common Mistakes to Avoid

When solving absolute value equations, it is important to avoid common mistakes that can lead to incorrect solutions. One frequent error is forgetting to consider both positive and negative cases for the expression inside the absolute value. Failing to account for both possibilities will result in missing one of the solutions. Another common mistake is incorrectly manipulating the equation when dealing with fractions. It's crucial to perform algebraic operations carefully, ensuring that each step is logically sound and maintains the equality of the equation. Additionally, always verify your solutions by plugging them back into the original equation. This step helps to identify any errors made during the solving process and confirms the accuracy of the results. By being mindful of these potential pitfalls and consistently applying the correct techniques, you can improve your ability to solve absolute value equations accurately and efficiently.

Conclusion

In conclusion, we have successfully solved the absolute value equation $\left|\frac{5}{x-5}\right|=15$. By carefully considering both positive and negative cases and applying algebraic techniques, we found the solutions to be $x = \frac{16}{3}$ and $x = \frac{14}{3}$. We also verified these solutions to ensure their accuracy. Mastering the process of solving absolute value equations is essential for a strong foundation in algebra. Remember to always consider both cases and verify your solutions to avoid common mistakes. With practice, you can confidently tackle a wide range of absolute value equations. This step-by-step approach not only helps in solving this specific equation but also provides a framework for tackling similar problems in the future.