Solving the Equation 5/x - 3/x -8: A Comprehensive Guide

Mathematical equations can sometimes seem daunting, but with the right techniques and understanding, they can be easily solved. In this article, we will guide you through the process of solving the equation 5/x - 3/x -8. By the end, you will understand the step-by-step solution and be able to tackle similar problems on your own.

Understanding the Premise: 5/x - 3/x -8

When we face a problem like 5/x - 3/x -8, the first step is to understand the premise. The equation involves rational expressions where the numerators (5 and 3) are being subtracted from one another, while the denominators (x) are the same. The goal is to find the value of x that satisfies the equation.

Step-by-Step Solution

Step 1:
Start by simplifying the left side of the equation:

5/x - 3/x -8

When we subtract these fractions, the denominator becomes the common term:

5 - 3/x -8

Which simplifies to:

2/x -8

Step 2: Isolating the Denominator

The next step is to isolate the denominator on one side of the equation. We can do this by multiplying both sides by x:

(2/x) * x -8 * x

This simplifies to:

2 -8x

Step 3: Solving for x

To solve for x, we divide both sides of the equation by -8:

2 / (-8) x

This simplifies to:

x -1/4

x -0.25

Thus, the solution to the equation 5/x - 3/x -8 is x -1/4, or -0.25.

Verification

To verify the solution, we substitute x -1/4 back into the original equation:

5/(-1/4) - 3/(-1/4) -8

This can be further simplified as follows:

5 * (-4) - 3 * (-4) -8

The equation becomes:

-20 - (-12) -8

-20 12 -8

-8 -8

This confirms that the solution is correct.

Conclusion

By following each step carefully and understanding the process, we can solve the equation 5/x - 3/x -8. The final answer is x -1/4, or in decimal form, x -0.25. This method can be applied to other similar equations, making it a valuable skill in algebra and mathematical problem-solving.