Solving The Equation: 1/x + 1/(x+4) = 1/6 - A Step-by-Step Guide
Hey guys! Today, we're diving into a common yet sometimes tricky math problem: solving equations with fractions. Specifically, we're going to break down how to solve the equation 1/x + 1/(x+4) = 1/6. This type of problem often pops up in algebra, and mastering it is super useful for more advanced math. So, grab your thinking caps, and let's get started!
Understanding the Equation
Before we jump into solving, let's make sure we understand what the equation 1/x + 1/(x+4) = 1/6 actually means. We're looking for the value(s) of 'x' that will make this equation true. The equation involves fractions with 'x' in the denominator, which means we need to be careful about values of 'x' that would make the denominator zero (since division by zero is a big no-no in math!). This means x cannot be 0 or -4. Keep that in mind as we work through the solution. Remember, understanding the constraints is just as crucial as finding the solution itself.
The core of solving this equation lies in manipulating it algebraically to isolate 'x'. We'll be using several key mathematical principles, including finding a common denominator, simplifying fractions, and solving quadratic equations. Don't worry if some of these terms sound intimidating; we'll break each step down so it’s easy to follow. Throughout this process, we aim to maintain the equation's balance – whatever we do on one side, we must do on the other. This is the golden rule of equation solving, ensuring that we arrive at the correct solution without altering the fundamental relationship expressed by the equation.
Why is this important? Well, equations like this pop up in all sorts of real-world scenarios, from physics and engineering to finance and computer science. Being able to confidently solve them is a valuable skill. It's not just about getting the right answer; it's about understanding the process and being able to apply it to new problems. Think of it as building a muscle – the more you practice, the stronger your problem-solving abilities become. So, let's flex those mathematical muscles and tackle this equation together!
Step 1: Finding a Common Denominator
The first step in solving this equation is to combine the fractions on the left side. To do this, we need to find a common denominator. In this case, the common denominator for 'x' and 'x+4' is simply their product: 'x(x+4)'.
Think of it like adding fractions with different denominators, like 1/2 + 1/3. You can't directly add them until you find a common denominator (which would be 6 in that case). We're doing the same thing here, but with expressions instead of numbers. So, we'll rewrite each fraction with the denominator 'x(x+4)'.
To get the first fraction, 1/x, to have the denominator 'x(x+4)', we multiply both the numerator and denominator by '(x+4)'. This gives us (1 * (x+4)) / (x * (x+4)), which simplifies to (x+4) / x(x+4). Similarly, for the second fraction, 1/(x+4), we multiply both the numerator and denominator by 'x', resulting in (1 * x) / ((x+4) * x), which simplifies to x / x(x+4). Remember, multiplying the numerator and denominator by the same expression doesn't change the value of the fraction; it's like multiplying by 1.
Now our equation looks like this: (x+4) / x(x+4) + x / x(x+4) = 1/6. With a common denominator, we can now combine the fractions on the left side. This is a crucial step because it simplifies the equation and allows us to work with a single fraction instead of two. Finding the common denominator is often the key to unlocking these types of equations.
Step 2: Combining the Fractions
Now that we have a common denominator, we can combine the fractions on the left side of the equation. This is a straightforward step: we simply add the numerators while keeping the common denominator. So, (x+4) / x(x+4) + x / x(x+4) becomes (x + 4 + x) / x(x+4). This simplifies to (2x + 4) / x(x+4).
Our equation now looks like this: (2x + 4) / x(x+4) = 1/6. See how much simpler it's getting? By combining the fractions, we've reduced the complexity of the left side, making it easier to work with. This step highlights the power of algebraic manipulation in simplifying complex expressions.
At this point, it's worth pausing and appreciating the progress we've made. We started with an equation that looked a bit intimidating, with two separate fractions on one side. But by systematically applying the principles of fraction addition, we've transformed it into a single, more manageable fraction. This is a testament to the power of breaking down complex problems into smaller, more digestible steps. The next step involves getting rid of the fractions altogether, which will bring us even closer to solving for 'x'.
Step 3: Cross-Multiplication
To get rid of the fractions, we'll use a technique called cross-multiplication. This is a handy trick when you have an equation with a fraction on each side. The idea is to multiply the numerator of the left fraction by the denominator of the right fraction, and vice-versa.
In our case, we have (2x + 4) / x(x+4) = 1/6. Cross-multiplying gives us (2x + 4) * 6 = 1 * x(x+4). Let's simplify this: 6(2x + 4) = x(x+4). Now we have an equation without fractions, which is much easier to handle. Cross-multiplication is essentially a shortcut for multiplying both sides of the equation by the denominators, but it's a quick and efficient way to eliminate fractions in one step.
Remember, cross-multiplication is a valid technique because it preserves the equality of the equation. We're essentially multiplying both sides by the same quantities, which doesn't change the solution. However, it's important to use it correctly, multiplying the correct numerators and denominators. Cross-multiplication is a powerful tool, but it's crucial to understand why it works and to apply it accurately. In the next step, we'll simplify this new equation and move closer to finding the value(s) of 'x'.
Step 4: Simplifying and Rearranging
Now let's simplify the equation we got from cross-multiplication: 6(2x + 4) = x(x+4). First, we'll distribute on both sides. On the left side, 6 * 2x = 12x and 6 * 4 = 24, so we have 12x + 24. On the right side, x * x = x² and x * 4 = 4x, giving us x² + 4x.
Our equation now looks like this: 12x + 24 = x² + 4x. To solve for 'x', we want to get all the terms on one side, making the other side equal to zero. This will allow us to use techniques for solving quadratic equations. So, we'll subtract 12x and 24 from both sides. This gives us 0 = x² + 4x - 12x - 24, which simplifies to 0 = x² - 8x - 24.
We now have a quadratic equation in the standard form: ax² + bx + c = 0, where a = 1, b = -8, and c = -24. Rearranging the equation into this standard form is a crucial step in solving quadratic equations. It sets us up to use methods like factoring, completing the square, or the quadratic formula, which we'll explore in the next step. The key takeaway here is the importance of algebraic manipulation in transforming the equation into a form that's easier to solve.
Step 5: Solving the Quadratic Equation
We've arrived at the quadratic equation: x² - 8x - 24 = 0. There are a couple of ways we can solve this: factoring or using the quadratic formula. Let's start by trying to factor. Factoring involves finding two numbers that multiply to -24 (the constant term) and add up to -8 (the coefficient of the x term).
After trying a few combinations, we might realize that factoring this quadratic equation isn't straightforward. Sometimes, quadratic equations just don't factor nicely. That's where the quadratic formula comes in handy. The quadratic formula is a universal tool for solving any quadratic equation in the form ax² + bx + c = 0. It states that:
x = (-b ± √(b² - 4ac)) / 2a
In our case, a = 1, b = -8, and c = -24. Plugging these values into the formula, we get:
x = (8 ± √((-8)² - 4 * 1 * -24)) / (2 * 1)
Let's simplify this step by step. First, (-8)² = 64, and -4 * 1 * -24 = 96. So, we have:
x = (8 ± √(64 + 96)) / 2
Which simplifies to:
x = (8 ± √160) / 2
We can simplify √160 by factoring out the largest perfect square, which is 16. √160 = √(16 * 10) = 4√10. So, our equation becomes:
x = (8 ± 4√10) / 2
Finally, we can divide both terms in the numerator by 2:
x = 4 ± 2√10
This gives us two possible solutions for x: x = 4 + 2√10 and x = 4 - 2√10. The quadratic formula is a powerful tool because it guarantees a solution, even when factoring is difficult or impossible.
Step 6: Checking for Extraneous Solutions
We've found two possible solutions for x: x = 4 + 2√10 and x = 4 - 2√10. However, it's crucial to check these solutions to make sure they are valid. Remember, we had a restriction at the beginning: x cannot be 0 or -4 because those values would make the denominators in the original equation zero.
Let's think about what extraneous solutions are. Extraneous solutions are values that we get when solving an equation that don't actually satisfy the original equation. This often happens when we perform operations that can introduce new solutions, like squaring both sides of an equation or, in our case, dealing with rational expressions. Therefore, it’s always vital to verify our solutions against the initial equation’s constraints and conditions.
Our solutions, 4 + 2√10 and 4 - 2√10, are both real numbers. 4 + 2√10 is approximately 10.32, and 4 - 2√10 is approximately -2.32. Neither of these values is 0 or -4, so they don't violate our initial restriction. This means both solutions are valid. If we had found a solution that was 0 or -4, we would have had to discard it as an extraneous solution.
Conclusion
Alright, guys, we did it! We successfully solved the equation 1/x + 1/(x+4) = 1/6. We navigated through finding a common denominator, combining fractions, cross-multiplication, simplifying, and using the quadratic formula. We also remembered to check for extraneous solutions, which is a critical step in ensuring the accuracy of our results.
Our solutions are x = 4 + 2√10 and x = 4 - 2√10. This problem demonstrates the power of algebraic techniques in solving complex equations. Remember, the key is to break down the problem into smaller, manageable steps, apply the appropriate tools, and always double-check your work. Keep practicing, and you'll become a master equation solver in no time!
