Solving The Rational Equation (-10)/(q-2) - 7/(q+4) = 1 A Step-by-Step Guide
Introduction
In this comprehensive guide, we will delve into the process of solving the rational equation (-10)/(q-2) - 7/(q+4) = 1. Rational equations, characterized by fractions with variables in the denominator, often present a unique set of challenges. This article aims to provide a clear, step-by-step approach to tackling such equations, ensuring you not only arrive at the correct solution(s) but also understand the underlying principles involved. We'll explore the importance of identifying restricted values, finding a common denominator, and the critical step of verifying solutions to avoid extraneous roots. Whether you're a student grappling with algebra or simply looking to refresh your mathematical skills, this guide will equip you with the knowledge and confidence to solve rational equations effectively.
Understanding Rational Equations
Before diving into the specific equation at hand, let's first establish a firm understanding of rational equations. A rational equation is essentially an equation that contains at least one fraction whose numerator and/or denominator are polynomials. The presence of variables in the denominator introduces a crucial consideration: we must identify values that would make the denominator zero, as division by zero is undefined. These values are known as restricted values and must be excluded from the solution set. Furthermore, when working with rational equations, it's essential to manipulate the equation algebraically to eliminate the fractions, typically by multiplying both sides by the least common denominator (LCD). This process transforms the rational equation into a more manageable polynomial equation. However, it's crucial to remember that this manipulation might introduce extraneous solutions, which are values that satisfy the transformed equation but not the original rational equation. Therefore, verifying solutions by substituting them back into the original equation is a vital step in the process. Understanding these core concepts forms the foundation for successfully solving rational equations.
Identifying Restricted Values
The first crucial step in solving the rational equation (-10)/(q-2) - 7/(q+4) = 1 is to identify any restricted values. These are the values of q that would make any of the denominators in the equation equal to zero, rendering the expression undefined. In our equation, we have two denominators: (q-2) and (q+4). To find the restricted values, we set each denominator equal to zero and solve for q. For the first denominator, q-2 = 0, adding 2 to both sides gives us q = 2. For the second denominator, q+4 = 0, subtracting 4 from both sides gives us q = -4. Therefore, the restricted values for this equation are q = 2 and q = -4. These values must be excluded from our solution set. Identifying these restrictions upfront is paramount, as failing to do so can lead to the acceptance of extraneous solutions, which are not valid solutions to the original equation. By explicitly recognizing and noting these restricted values, we ensure that our solution process remains accurate and that the final answers are mathematically sound. Remember, the restricted values are critical checkpoints in the process of solving rational equations.
Finding the Least Common Denominator (LCD)
After identifying the restricted values, the next step in solving our rational equation (-10)/(q-2) - 7/(q+4) = 1 is to determine the least common denominator (LCD). The LCD is the smallest expression that is divisible by all the denominators in the equation. In this case, our denominators are (q-2) and (q+4). Since these are distinct linear expressions with no common factors, the LCD is simply their product: (q-2)(q+4). Finding the LCD is crucial because it allows us to eliminate the fractions in the equation, transforming it into a more manageable form. To do this, we will multiply both sides of the equation by the LCD. This process effectively clears the fractions, making the equation easier to solve. Understanding how to find the LCD is a fundamental skill in solving rational equations, as it simplifies the equation and paves the way for further algebraic manipulation. Once we have the LCD, we can proceed to multiply both sides of the equation by it, setting the stage for solving for the variable q.
Multiplying by the LCD and Simplifying
With the LCD identified as (q-2)(q+4), we now proceed to multiply both sides of the rational equation (-10)/(q-2) - 7/(q+4) = 1 by this expression. This crucial step eliminates the fractions, simplifying the equation into a more manageable form. When we multiply the left side by (q-2)(q+4), we distribute the LCD to each term. For the first term, (-10)/(q-2) multiplied by (q-2)(q+4) simplifies to -10(q+4), as the (q-2) factors cancel out. For the second term, -7/(q+4) multiplied by (q-2)(q+4) simplifies to -7(q-2), as the (q+4) factors cancel out. On the right side, 1 multiplied by (q-2)(q+4) remains as (q-2)(q+4). Thus, our equation now transforms into -10(q+4) - 7(q-2) = (q-2)(q+4). Next, we need to distribute and simplify both sides of the equation. Distributing on the left side, we get -10q - 40 - 7q + 14. Expanding the product on the right side, we obtain q² + 4q - 2q - 8, which simplifies to q² + 2q - 8. Combining like terms on the left side yields -17q - 26. Therefore, our equation is now -17q - 26 = q² + 2q - 8. This simplification process is vital in solving rational equations, as it transforms a complex-looking equation into a more familiar quadratic form.
Solving the Quadratic Equation
After simplifying the equation, we have -17q - 26 = q² + 2q - 8. To solve this quadratic equation, we need to rearrange it into the standard quadratic form, which is ax² + bx + c = 0. We can achieve this by adding 17q and 26 to both sides of the equation. This gives us 0 = q² + 2q + 17q - 8 + 26, which simplifies to q² + 19q + 18 = 0. Now that we have the equation in standard quadratic form, we can proceed to solve for q. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a viable option. We are looking for two numbers that multiply to 18 and add up to 19. These numbers are 1 and 18. Therefore, we can factor the quadratic expression as (q + 1)(q + 18) = 0. To find the solutions, we set each factor equal to zero. So, q + 1 = 0 gives us q = -1, and q + 18 = 0 gives us q = -18. Thus, we have obtained two potential solutions: q = -1 and q = -18. However, it is crucial to remember the restricted values we identified earlier. These values must be excluded from our final solution set. This step is crucial in solving rational equations.
Verifying the Solutions and Identifying Extraneous Roots
Having obtained potential solutions, the final critical step in solving the rational equation is to verify these solutions and identify any extraneous roots. Extraneous roots are values that satisfy the transformed equation (in this case, the quadratic equation) but do not satisfy the original rational equation. To verify our solutions, q = -1 and q = -18, we substitute each value back into the original equation: (-10)/(q-2) - 7/(q+4) = 1. Let's start with q = -1. Substituting -1 for q, we get (-10)/(-1-2) - 7/(-1+4) = (-10)/(-3) - 7/3 = 10/3 - 7/3 = 3/3 = 1. Since this holds true, q = -1 is a valid solution. Now, let's verify q = -18. Substituting -18 for q, we get (-10)/(-18-2) - 7/(-18+4) = (-10)/(-20) - 7/(-14) = 1/2 + 1/2 = 1. This also holds true, so q = -18 is also a valid solution. Importantly, neither of our solutions, q = -1 and q = -18, match the restricted values we identified earlier (q = 2 and q = -4). Therefore, neither solution is extraneous. This verification process is essential in solving rational equations, as it ensures that the solutions we obtain are indeed valid and not artifacts of the algebraic manipulations. In conclusion, after verifying both solutions, we can confidently state that the solutions to the equation (-10)/(q-2) - 7/(q+4) = 1 are q = -1 and q = -18.
Final Answer
After diligently following the steps of identifying restricted values, finding the least common denominator, multiplying to eliminate fractions, solving the resulting quadratic equation, and verifying the solutions, we have arrived at the final answer. The solutions to the rational equation (-10)/(q-2) - 7/(q+4) = 1 are q = -1 and q = -18. It's worth reiterating the importance of each step in this process. Identifying restricted values at the outset prevents the inclusion of extraneous solutions. Finding the LCD allows us to clear fractions, simplifying the equation. Solving the resulting quadratic equation provides potential solutions, and crucially, verifying these solutions against the original equation ensures their validity. By meticulously following these steps, we can confidently and accurately solve rational equations. Therefore, the solutions to the given equation are q = -1 and q = -18.
