Solving Rational Equations: A Step-by-Step Guide

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Let's dive into how to solve the rational equation mm+4+44−m=m2m2−16\frac{m}{m+4}+\frac{4}{4-m}=\frac{m^2}{m^2-16}. This type of problem involves fractions with variables in the denominator, which can seem a bit daunting at first. But don't worry, we'll break it down into manageable steps so you can tackle it with confidence. Solving rational equations is a fundamental skill in algebra, with applications in various fields like physics, engineering, and economics. Understanding how to manipulate and solve these equations will not only help you in your math courses but also provide you with valuable problem-solving tools for real-world scenarios. So, grab your pencil and paper, and let's get started! We'll go through each step meticulously, ensuring you grasp the underlying concepts and can apply them to similar problems. Remember, practice makes perfect, so the more you work with these types of equations, the easier they'll become. Let's begin by identifying the domain restrictions, a crucial first step in solving rational equations.

1. Identifying Domain Restrictions

First, it's crucial to identify any values of m that would make the denominators equal to zero, as division by zero is undefined. Looking at the equation mm+4+44−m=m2m2−16\frac{m}{m+4}+\frac{4}{4-m}=\frac{m^2}{m^2-16}, we have three denominators to consider: m + 4, 4 - m, and m2 - 16. Why is this important, guys? Because if any of these denominators equal zero, the equation becomes undefined, and those values of m cannot be solutions. So, let's find those values.

  • For m + 4 = 0, we get m = -4.
  • For 4 - m = 0, we get m = 4.
  • For m2 - 16 = 0, we can factor this as (m + 4)(m - 4) = 0, which gives us m = -4 and m = 4. Notice that m2-16 can be factored using the difference of squares. Factoring is a super handy technique, so keep it in your toolkit! These are the values of m that we must exclude from our possible solutions. So, m cannot be 4 or -4. Keep this in mind as we proceed; if we end up with either of these values as a potential solution, we'll have to discard it. Identifying domain restrictions is a critical first step because it prevents us from accepting extraneous solutions that might arise during the algebraic manipulation of the equation. Ignoring these restrictions can lead to incorrect answers, so always make it a habit to check for them at the beginning of every problem involving rational expressions.

2. Finding a Common Denominator

Now that we know the restrictions on m, let's find a common denominator for the fractions in the equation. We have mm+4+44−m=m2m2−16\frac{m}{m+4}+\frac{4}{4-m}=\frac{m^2}{m^2-16}. Notice that m2 - 16 can be factored into (m + 4)(m - 4). Also, notice that 4 - m is the negative of m - 4. So, we can rewrite the equation as follows:

mm+4−4m−4=m2(m+4)(m−4)\frac{m}{m+4}-\frac{4}{m-4}=\frac{m^2}{(m+4)(m-4)}

We changed the plus sign to a minus sign because we factored out a -1. Factoring and recognizing patterns are key skills here, guys! The common denominator, therefore, is (m + 4)(m - 4). This common denominator allows us to combine the fractions on the left side of the equation into a single fraction. By expressing each term with the same denominator, we can then equate the numerators and solve for the variable m. This simplifies the equation and makes it easier to manipulate algebraically. Finding a common denominator is a standard technique when dealing with addition or subtraction of fractions, and it's particularly useful in the context of rational equations. It allows us to combine multiple terms into a single expression, which can then be further simplified or solved. Remember to always look for opportunities to factor expressions and simplify the equation before proceeding with finding the common denominator. This can save you time and effort in the long run.

3. Rewriting the Equation with the Common Denominator

Let's rewrite each term with the common denominator (m + 4)(m - 4):

m(m−4)(m+4)(m−4)−4(m+4)(m−4)(m+4)=m2(m+4)(m−4)\frac{m(m-4)}{(m+4)(m-4)}-\frac{4(m+4)}{(m-4)(m+4)}=\frac{m^2}{(m+4)(m-4)}

Now that all the fractions have the same denominator, we can combine the terms on the left side:

m(m−4)−4(m+4)(m+4)(m−4)=m2(m+4)(m−4)\frac{m(m-4)-4(m+4)}{(m+4)(m-4)}=\frac{m^2}{(m+4)(m-4)}

This step is crucial because it sets the stage for eliminating the denominators and simplifying the equation into a more manageable form. By rewriting each term with the common denominator, we ensure that we are working with equivalent expressions that can be combined without changing the solution set. This process involves multiplying the numerator and denominator of each fraction by the appropriate factors to obtain the desired common denominator. It's important to pay close attention to the signs and ensure that the expressions are correctly manipulated. Once all the terms have the same denominator, we can proceed to combine the numerators and simplify the equation further. Remember, the goal is to transform the equation into a form that is easier to solve, and rewriting with the common denominator is a key step in achieving that goal. It's also a good practice to double-check your work to ensure that you haven't made any errors in the process, as even a small mistake can lead to an incorrect solution.

4. Simplifying the Equation

Now, let's simplify the numerator on the left side:

m2−4m−4m−16(m+4)(m−4)=m2(m+4)(m−4)\frac{m^2-4m-4m-16}{(m+4)(m-4)}=\frac{m^2}{(m+4)(m-4)}

m2−8m−16(m+4)(m−4)=m2(m+4)(m−4)\frac{m^2-8m-16}{(m+4)(m-4)}=\frac{m^2}{(m+4)(m-4)}

Since the denominators are the same on both sides, we can eliminate them (as long as m≠4m \neq 4 and m≠−4m \neq -4). This gives us:

m2−8m−16=m2m^2-8m-16=m^2

This step is valid only because we've already identified the restrictions on m. Simplifying the equation is a critical step in solving for the unknown variable. By combining like terms, canceling out common factors, and eliminating denominators, we can reduce the equation to its simplest form, making it easier to isolate the variable and find its value. In this case, simplifying the numerator on the left side involves expanding the terms and combining like terms. Then, since the denominators are the same on both sides of the equation, we can eliminate them, which simplifies the equation even further. However, it's important to remember that this step is valid only because we've already identified the restrictions on m. If we hadn't done that, we might inadvertently introduce extraneous solutions. Simplifying the equation not only makes it easier to solve but also reduces the chances of making errors in the subsequent steps. It's always a good idea to take the time to simplify the equation as much as possible before proceeding with solving for the variable. This will save you time and effort in the long run and increase your chances of finding the correct solution.

5. Solving for m

Subtracting m2 from both sides, we get:

-8m - 16 = 0

Adding 16 to both sides:

-8m = 16

Dividing by -8:

m = -2

So, our potential solution is m = -2. This step involves isolating the variable m on one side of the equation to determine its value. By performing algebraic operations such as adding, subtracting, multiplying, and dividing both sides of the equation by the same quantity, we can manipulate the equation until the variable is isolated. In this case, we subtract m2 from both sides to eliminate the m2 term, then add 16 to both sides to isolate the term with m. Finally, we divide both sides by -8 to solve for m. The result is m = -2, which is our potential solution. However, it's important to remember that we need to check whether this solution satisfies the original equation and whether it violates any domain restrictions. Solving for the variable is the ultimate goal of solving the equation, but it's not the end of the process. We still need to verify the solution and ensure that it is valid. Remember to always double-check your work and ensure that you haven't made any errors in the algebraic manipulations.

6. Checking the Solution

We found that m = -2. Recall that m cannot be 4 or -4. Since -2 is not equal to 4 or -4, it is a valid solution. But let's be absolutely sure and plug it back into the original equation: Check your solutions by substituting them back into the original equation to make sure they work.

−2−2+4+44−(−2)=(−2)2(−2)2−16\frac{-2}{-2+4}+ \frac{4}{4-(-2)} = \frac{(-2)^2}{(-2)^2-16}

−22+46=44−16\frac{-2}{2} + \frac{4}{6} = \frac{4}{4-16}

−1+23=4−12-1 + \frac{2}{3} = \frac{4}{-12}

−33+23=−13\frac{-3}{3} + \frac{2}{3} = \frac{-1}{3}

−13=−13\frac{-1}{3} = \frac{-1}{3}

The solution m = -2 checks out! Woo-hoo! Checking the solution is a crucial step in the process of solving equations. It involves substituting the value(s) obtained for the variable back into the original equation to verify that they satisfy the equation. This step helps to identify any extraneous solutions that may have arisen during the algebraic manipulation of the equation. Extraneous solutions are values that satisfy the simplified equation but not the original equation. In the case of rational equations, extraneous solutions often occur when we eliminate denominators or take square roots. By checking the solution, we can ensure that it is a valid solution and that it satisfies all the conditions of the original equation. If the solution does not check out, it means that we have made an error somewhere in the process, and we need to go back and review our work. Checking the solution is a simple but effective way to catch errors and ensure that we have found the correct solution. It's always a good idea to make it a habit to check your solutions, especially when dealing with rational equations or other types of equations where extraneous solutions are possible.

Final Answer

Therefore, the solution to the equation mm+4+44−m=m2m2−16\frac{m}{m+4}+\frac{4}{4-m}=\frac{m^2}{m^2-16} is m = -2. So the answer is D. Nice job, guys! You've successfully navigated the world of rational equations. Remember, the key is to take it one step at a time, and always double-check your work. You got this!