Solving Rational Equations A Step-by-Step Guide To Solve (x-2)/(x+3) - 1 = 3/(x+2)
In this article, we will delve into the step-by-step process of solving the algebraic equation (x-2)/(x+3) - 1 = 3/(x+2). This equation involves rational expressions, and our goal is to find the value(s) of x that satisfy the equation. We will explore the techniques required to manipulate the equation, clear the fractions, and ultimately arrive at the solution(s). This comprehensive guide is designed to help students, educators, and anyone interested in enhancing their algebraic problem-solving skills. Let’s embark on this journey to unravel the mysteries of this equation and gain a deeper understanding of the underlying mathematical concepts.
Step 1: Combining Terms on the Left-Hand Side
The first step in solving the equation (x-2)/(x+3) - 1 = 3/(x+2) involves simplifying the left-hand side (LHS) by combining the terms. We begin by expressing the number 1 as a fraction with the same denominator as the first term, which is (x+3). This allows us to subtract the terms effectively. Rewriting 1 as (x+3)/(x+3), the equation becomes:
(x-2)/(x+3) - (x+3)/(x+3) = 3/(x+2)
Now, we can combine the numerators over the common denominator:
[(x-2) - (x+3)] / (x+3) = 3/(x+2)
Simplifying the numerator, we have:
(x - 2 - x - 3) / (x+3) = 3/(x+2)
Which further simplifies to:
-5 / (x+3) = 3/(x+2)
This simplified equation sets the stage for the next step, where we will clear the fractions to facilitate solving for x. By combining terms and simplifying the LHS, we have made the equation more manageable and paved the way for further algebraic manipulation.
Step 2: Clearing the Fractions
To eliminate the fractions in the equation -5 / (x+3) = 3 / (x+2), we will multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is (x+3)(x+2). This process, known as clearing fractions, transforms the equation into a more easily solvable form. Multiplying both sides by (x+3)(x+2) gives us:
(x+3)(x+2) * [-5 / (x+3)] = (x+3)(x+2) * [3 / (x+2)]
On the left-hand side, the (x+3) terms cancel out, and on the right-hand side, the (x+2) terms cancel out. This leaves us with:
-5(x+2) = 3(x+3)
Now, we have a linear equation without fractions, which is much simpler to solve. The next step involves expanding the expressions on both sides and gathering like terms to isolate the variable x. Clearing the fractions has significantly simplified the equation, making it easier to find the value(s) of x that satisfy the original equation. This step is crucial in solving rational equations, as it removes the complexities introduced by the fractions and allows us to proceed with standard algebraic techniques.
Step 3: Expanding and Simplifying
After clearing the fractions in the equation, we arrived at the linear equation -5(x+2) = 3(x+3). The next step is to expand both sides of the equation by distributing the constants. Expanding the left-hand side, we have:
-5 * x + (-5) * 2 = -5x - 10
Expanding the right-hand side, we have:
3 * x + 3 * 3 = 3x + 9
So the equation becomes:
-5x - 10 = 3x + 9
Now, we need to simplify the equation by gathering like terms. This involves moving all terms containing x to one side of the equation and all constant terms to the other side. To do this, we can add 5x to both sides:
-5x - 10 + 5x = 3x + 9 + 5x
Which simplifies to:
-10 = 8x + 9
Next, we subtract 9 from both sides to isolate the term with x:
-10 - 9 = 8x + 9 - 9
This simplifies to:
-19 = 8x
Now, the equation is simplified to a form where we can easily isolate x by dividing both sides by the coefficient of x. This step of expanding and simplifying is crucial for solving linear equations, as it organizes the terms and makes it easier to isolate the variable. By carefully distributing, combining like terms, and rearranging the equation, we have brought ourselves closer to the solution.
Step 4: Isolating x
Following the simplification process, we have the equation -19 = 8x. To isolate x, we need to divide both sides of the equation by the coefficient of x, which is 8. This will give us the value of x that satisfies the equation. Dividing both sides by 8, we get:
-19 / 8 = (8x) / 8
This simplifies to:
x = -19/8
So, the solution to the equation is x = -19/8. This means that when x is equal to -19/8, the original equation (x-2)/(x+3) - 1 = 3/(x+2) holds true. Isolating x is the final step in solving a linear equation, and it involves performing the necessary operations to get x by itself on one side of the equation. By dividing both sides by the coefficient of x, we have successfully found the value of x that makes the equation true.
Step 5: Checking for Extraneous Solutions
In solving rational equations, it is crucial to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original equation. These solutions can arise when we clear fractions, as this process can sometimes introduce values that make the denominators in the original equation equal to zero, which is undefined.
Our original equation was (x-2)/(x+3) - 1 = 3/(x+2), and we found the solution x = -19/8. To check for extraneous solutions, we need to substitute this value back into the original equation and ensure that it does not make any of the denominators zero. The denominators in the original equation are (x+3) and (x+2). Let's substitute x = -19/8 into these denominators:
For (x+3):
(-19/8) + 3 = (-19/8) + (24/8) = 5/8
For (x+2):
(-19/8) + 2 = (-19/8) + (16/8) = -3/8
Since neither of the denominators is zero when x = -19/8, this solution is not extraneous. Therefore, x = -19/8 is a valid solution to the original equation. Checking for extraneous solutions is a critical step in solving rational equations, as it ensures that the solutions we find are valid and do not lead to undefined expressions in the original equation. By substituting the solution back into the original equation and verifying that it does not make any denominators zero, we can be confident in the accuracy of our solution.
Conclusion
In this comprehensive guide, we have successfully navigated the process of solving the equation (x-2)/(x+3) - 1 = 3/(x+2). We began by combining terms on the left-hand side, then cleared the fractions by multiplying both sides by the least common multiple of the denominators. Next, we expanded and simplified the resulting equation, isolated x, and arrived at the solution x = -19/8. Finally, we checked for extraneous solutions by substituting the solution back into the original equation and verifying that it did not make any denominators zero.
This step-by-step approach not only provides the solution to the specific equation but also offers valuable insights into the techniques used to solve rational equations in general. By understanding the principles of combining terms, clearing fractions, simplifying expressions, and checking for extraneous solutions, readers can enhance their algebraic problem-solving skills and tackle a wide range of similar equations with confidence. This guide serves as a valuable resource for students, educators, and anyone seeking to deepen their understanding of algebraic concepts and techniques. The journey through this equation has highlighted the importance of methodical problem-solving and the satisfaction of arriving at a correct solution through careful and deliberate steps.
