Solving Rational Equations: A Step-by-Step Guide

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Solving Rational Equations: A Step-by-Step Guide

Hey math enthusiasts! Ever stumbled upon a rational equation and felt a bit lost? Don't worry, it's totally normal. Rational equations might seem intimidating at first glance, but with the right approach, they become quite manageable. In this article, we'll dive deep into solving rational equations, breaking down each step with clarity and providing you with the tools you need to conquer these problems. We'll be working through the specific equation: $\frac{5}{x+1}-\frac{5}{2}=\frac{6}{3 x+3}$. So, grab your pencils, and let's get started!

Understanding Rational Equations

Okay, before we jump into the nitty-gritty of solving, let's make sure we're all on the same page about what a rational equation actually is. Basically, it's an equation that contains one or more rational expressions. And what's a rational expression? Well, it's just a fraction where the numerator and/or the denominator are polynomials. So, think of things like x+2xβˆ’1\frac{x+2}{x-1}, 3x2+4\frac{3}{x^2+4}, or even the equation we're about to tackle: 5x+1βˆ’52=63x+3\frac{5}{x+1}-\frac{5}{2}=\frac{6}{3 x+3}. See, not so scary, right? The key thing to remember is that you'll have variables in the denominators of your fractions. This is super important because it brings with it the potential for undefined values – those values of x that would make the denominator equal to zero. We'll need to keep an eye out for those when we solve, because they can become extraneous solutions (solutions that don't actually work in the original equation).

When we're tackling rational equations, the name of the game is usually to get rid of those pesky fractions. This is generally achieved by finding the least common denominator (LCD) of all the rational expressions in the equation. The LCD is the smallest expression that all the denominators divide into evenly. Once we have the LCD, we multiply every term in the equation by it. This cleverly cancels out the denominators, leaving us with a much simpler equation to solve (usually a linear or quadratic equation). After we solve the new equation, it's essential to check our solutions to make sure they're not extraneous. Let's start with our equation $\frac{5}{x+1}-\frac{5}{2}=\frac{6}{3 x+3}$ and work through the steps together. Are you ready to dive in, guys?

Step-by-Step Solution: Cracking the Code

Alright, let's roll up our sleeves and solve the equation: $\frac{5}{x+1}-\frac{5}{2}=\frac{6}{3 x+3}$. This is where the magic happens! Here's a breakdown of how to solve it, step by step, so you can follow along easily:

Step 1: Identify the Denominators and Find the LCD. First things first, we need to spot all the denominators in our equation. We've got x + 1, 2, and 3x + 3. Notice that 3x + 3 can be factored as 3(x + 1). This is super helpful! Now, the LCD has to include all the unique factors from our denominators, raised to the highest power they appear in any single denominator. In this case, the unique factors are 2 and (x + 1). Therefore, the LCD is 2(x + 1).

Step 2: Multiply Each Term by the LCD. This is where we clear the fractions. We multiply every term in the equation by 2(x + 1). Be meticulous here, because missing a term is a common mistake. Let's do it:

2(x+1)βˆ—5x+1βˆ’2(x+1)βˆ—52=2(x+1)βˆ—63(x+1)2(x+1) * \frac{5}{x+1} - 2(x+1) * \frac{5}{2} = 2(x+1) * \frac{6}{3(x+1)}

Notice how the (x + 1) in the first term cancels with the (x + 1) in the denominator, and the 2 in the second term cancels with the 2 in the denominator. The (x+1) in the third term cancels with (x+1) in the denominator, and 2 cancels out with 3. After simplification we get:

2βˆ—5βˆ’(x+1)βˆ—5=2βˆ—632*5 - (x+1)*5 = \frac{2*6}{3}

Step 3: Simplify and Solve the Resulting Equation. Now we simplify the equation, expand and collect like terms. This should leave us with something much easier to handle. So, let's simplify our equation:

10βˆ’5(x+1)=410 - 5(x+1) = 4

Expand the brackets:

10βˆ’5xβˆ’5=410 - 5x - 5 = 4

Combine like terms:

5βˆ’5x=45 - 5x = 4

Isolate the variable term:

βˆ’5x=βˆ’1-5x = -1

Solve for x:

x=15x = \frac{1}{5}

Step 4: Check for Extraneous Solutions. This step is CRUCIAL! Remember how we talked about values of x that make the denominator equal to zero? We need to make sure our solution, x = 1/5, doesn't fall into this category. Looking back at our original equation, the denominators are x + 1 and 3x + 3. Setting these equal to zero, we get x = -1. Our solution, x = 1/5, does not equal -1. Thus, x = 1/5 is not an extraneous solution.

Step 5: Verify the Solution. We can verify the solution by plugging x = 1/5 into the original equation: $\frac{5}{x+1}-\frac{5}{2}=\frac{6}{3 x+3}$. We get the following:

515+1βˆ’52=63βˆ—15+3\frac{5}{\frac{1}{5}+1}-\frac{5}{2}=\frac{6}{3 * \frac{1}{5}+3}

Simplifying each side:

565βˆ’52=6185\frac{5}{\frac{6}{5}}-\frac{5}{2}=\frac{6}{\frac{18}{5}}

Simplifying each side:

256βˆ’52=3018\frac{25}{6}-\frac{5}{2}=\frac{30}{18}

Simplifying each side:

256βˆ’156=53\frac{25}{6}-\frac{15}{6}=\frac{5}{3}

Simplifying each side:

106=53\frac{10}{6}=\frac{5}{3}

53=53\frac{5}{3}=\frac{5}{3}

Since both sides are equal, the solution is correct.

So, the solution to our rational equation is x = 1/5. Boom! You've successfully solved your first rational equation.

Tips for Success

Alright, you've seen how it's done. But, like anything in math, practice makes perfect. Here are some tips to help you become a rational equation-solving ninja:

  • Practice, Practice, Practice: The more you solve these problems, the more comfortable you'll become with the steps and the better you'll get at spotting potential pitfalls. Work through a variety of examples.
  • Always Check Your Solutions: Seriously, don't skip this step. It's easy to make a mistake when solving the simplified equation, and checking your answer will catch any errors.
  • Simplify, Simplify, Simplify: Before you start multiplying by the LCD, look for ways to simplify the rational expressions. This can make the entire process easier.
  • Factor, Factor, Factor: Factoring is your friend. It'll help you find the LCD and identify those pesky values of x that make the denominator zero.
  • Don't Be Afraid to Ask for Help: If you get stuck, don't suffer in silence. Ask your teacher, a friend, or use online resources for help. Understanding is way more important than struggling alone.

Conclusion: You've Got This!

Solving rational equations might seem like a climb, but with persistence, you'll reach the summit. Remember to follow the steps, be careful with your algebra, and always check your solutions. You’ve now got a solid foundation for tackling any rational equation that comes your way. Keep practicing, and you'll be solving these equations with confidence in no time! Keep up the great work, and happy calculating, guys! You've totally got this!