Solving (1/(x-1)) - (1/(x+5)) = 6/7: A Step-by-Step Guide

by Pedro Alvarez 58 views

Hey guys! Today, we're diving deep into the fascinating world of mathematics to tackle a specific equation: $\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}$. This might look a bit intimidating at first glance, but don't worry! We're going to break it down step by step, making sure everyone understands the process. Think of it like solving a puzzle – each step gets us closer to the final answer. Our journey will involve clearing fractions, simplifying expressions, and ultimately finding the values of 'x' that make this equation true. So, buckle up, grab your thinking caps, and let's get started on this mathematical adventure!

Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation actually means. We have two fractions on the left side, $\frac{1}{x-1}$ and $\frac{1}{x+5}$, being subtracted from each other. The result of this subtraction is equal to the fraction $ rac{6}{7}$. Our goal is to find the value(s) of 'x' that satisfy this relationship. In simpler terms, we're looking for the number(s) that, when plugged in for 'x', will make the left side of the equation equal to the right side. This is the core concept of solving equations, and it's a fundamental skill in algebra and beyond. We need to keep in mind that there might be restrictions on the values 'x' can take. For instance, if 'x' were to be 1, the first fraction would have a denominator of zero, which is undefined in mathematics. Similarly, if 'x' were -5, the second fraction would be undefined. These values are called undefined points, and we'll need to be mindful of them when we find our solutions.

Think of these restrictions like roadblocks on our path to the solution. We need to navigate around them to reach our destination. This careful consideration of potential pitfalls is what makes solving equations a thoughtful and rewarding process. We're not just blindly applying rules; we're actively thinking about the implications of each step. So, with a clear understanding of the equation and the potential roadblocks ahead, let's move on to the next step: clearing those pesky fractions!

Clearing the Fractions

Fractions can sometimes make equations look more complicated than they actually are. So, the first step in solving this equation is to get rid of the fractions. How do we do that? We use a technique called finding the least common denominator (LCD). The LCD is the smallest multiple that all the denominators in the equation share. In our case, the denominators are (x-1), (x+5), and 7. Since these expressions don't share any common factors, the LCD is simply their product: 7(x-1)(x+5). This LCD acts like a magic key that unlocks the equation and makes it easier to handle.

Now, we multiply both sides of the equation by the LCD. This is a crucial step, and it's important to do it carefully. We're essentially multiplying each term in the equation by 7(x-1)(x+5). When we do this, the denominators will cancel out, leaving us with a much simpler equation to work with. Let's break it down:

  • Multiplying the first term, $\frac{1}{x-1}$, by 7(x-1)(x+5), the (x-1) terms cancel, leaving us with 7(x+5).
  • Multiplying the second term, $\frac{1}{x+5}$, by 7(x-1)(x+5), the (x+5) terms cancel, leaving us with 7(x-1).
  • Multiplying the right side, $\frac{6}{7}$, by 7(x-1)(x+5), the 7s cancel, leaving us with 6(x-1)(x+5).

By performing this multiplication, we've transformed the equation from one involving fractions to a more manageable one involving only polynomials. This is a significant step forward, and it demonstrates the power of using the LCD to simplify equations. Now, we have a new equation: 7(x+5) - 7(x-1) = 6(x-1)(x+5). It looks much cleaner, doesn't it? We're now ready to move on to the next phase: simplifying this new equation.

Simplifying the Equation

Now that we've cleared the fractions, it's time to simplify the equation we obtained: 7(x+5) - 7(x-1) = 6(x-1)(x+5). This involves expanding the expressions and combining like terms. Think of it like tidying up a room – we're organizing the terms to make them easier to work with. First, let's expand the expressions on both sides of the equation. Remember the distributive property? We'll need to use it here. On the left side, we have:

  • 7(x+5) = 7x + 35
  • -7(x-1) = -7x + 7

Combining these, we get 7x + 35 - 7x + 7. Notice that the 7x and -7x terms cancel each other out! This is a nice simplification, leaving us with just 35 + 7 = 42 on the left side. On the right side, we have 6(x-1)(x+5). Let's expand the product (x-1)(x+5) first:

  • (x-1)(x+5) = x^2 + 5x - x - 5 = x^2 + 4x - 5

Now, we multiply this by 6: 6(x^2 + 4x - 5) = 6x^2 + 24x - 30. So, our equation now looks like this: 42 = 6x^2 + 24x - 30. We've made significant progress! The equation is now a quadratic equation, which we know how to solve. To make things even clearer, let's move all the terms to one side to set the equation equal to zero. Subtracting 42 from both sides, we get: 0 = 6x^2 + 24x - 72. This is a standard quadratic equation form (ax^2 + bx + c = 0), and we're ready to apply our quadratic equation-solving skills.

Simplifying the equation has transformed it into a familiar form, making it much easier to tackle. We've gone from a complex-looking equation with fractions to a clean and manageable quadratic equation. Now, we're just a couple of steps away from finding the solutions!

Solving the Quadratic Equation

We've arrived at the heart of the problem: solving the quadratic equation 6x^2 + 24x - 72 = 0. There are a couple of ways we can approach this. One common method is to use the quadratic formula, but before we jump into that, let's see if we can simplify the equation further. Notice that all the coefficients (6, 24, and -72) are divisible by 6. This means we can divide the entire equation by 6 to make the numbers smaller and the equation easier to handle. Dividing both sides by 6, we get: x^2 + 4x - 12 = 0. This is the same equation, just with smaller coefficients. It's much friendlier to work with!

Now, let's try to factor this quadratic equation. Factoring involves finding two binomials that, when multiplied together, give us the quadratic expression. We're looking for two numbers that multiply to -12 and add up to 4. After a little thought, we can see that the numbers 6 and -2 fit the bill. So, we can factor the equation as follows: (x + 6)(x - 2) = 0. Factoring is like reverse-engineering the multiplication process. We're breaking down the quadratic expression into its building blocks.

Now, here's the crucial step: if the product of two factors is zero, then at least one of the factors must be zero. This is known as the zero-product property. So, we have two possibilities:

  • x + 6 = 0
  • x - 2 = 0

Solving the first equation, x + 6 = 0, we subtract 6 from both sides and get x = -6. Solving the second equation, x - 2 = 0, we add 2 to both sides and get x = 2. So, we have two potential solutions: x = -6 and x = 2. We've done it! We've found the values of 'x' that make the quadratic equation true. But our journey isn't quite over yet. We need to make sure these solutions are valid in the original equation.

Checking the Solutions

We've found two potential solutions to our equation: x = -6 and x = 2. But before we declare victory, we need to check if these solutions actually work in the original equation: $\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}$. This is a crucial step in the problem-solving process. Think of it as double-checking your work to make sure you haven't made any mistakes along the way. Let's start with x = -6. We'll substitute -6 for 'x' in the original equation and see if the left side equals the right side:

  • 1−6−1−1−6+5=1−7−1−1=−17+1=67\frac{1}{-6-1}-\frac{1}{-6+5} = \frac{1}{-7} - \frac{1}{-1} = -\frac{1}{7} + 1 = \frac{6}{7}

Great! When x = -6, the left side of the equation equals the right side. So, x = -6 is a valid solution. Now, let's check x = 2:

  • 12−1−12+5=11−17=1−17=67\frac{1}{2-1}-\frac{1}{2+5} = \frac{1}{1} - \frac{1}{7} = 1 - \frac{1}{7} = \frac{6}{7}

Excellent! When x = 2, the left side of the equation also equals the right side. So, x = 2 is also a valid solution. We've successfully checked both solutions, and they both work! This confirms that our hard work has paid off. We've not only found the solutions but also verified that they are correct. Checking our solutions is like putting the final piece in the puzzle, completing the picture and giving us confidence in our answer.

Conclusion

Wow, we've made it! We successfully solved the equation $\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7}$. We started by understanding the equation, then cleared the fractions, simplified the resulting equation, solved the quadratic equation, and finally, checked our solutions. It was quite a journey, but we navigated it together, step by step. The solutions we found are x = -6 and x = 2. These are the values of 'x' that make the original equation true.

This problem demonstrates the power of breaking down complex problems into smaller, more manageable steps. We used several key mathematical concepts along the way, including finding the least common denominator, the distributive property, factoring quadratic equations, and the zero-product property. By understanding and applying these concepts, we were able to conquer the equation and find the solutions. More importantly, we learned a valuable problem-solving strategy that can be applied to many different situations. So, the next time you encounter a challenging problem, remember the steps we took today: understand, simplify, solve, and check. And remember, math can be an exciting adventure when you approach it with curiosity and a willingness to learn!