Consider The Following Rational Equation With Denominators Containing Variables: 1 X − 2 − 5 X + 5 = 7 X 2 + 3 X − 10 \frac{1}{x-2} - \frac{5}{x+5} = \frac{7}{x^2+3x-10} X − 2 1 − X + 5 5 = X 2 + 3 X − 10 7 A. What Is/are The Value(s) Of The Variable That Make The Denominators Zero? These Are The Restrictions On The

Mar 11, 2025 by ADMIN 324 views

**Solving Rational Equations with Variable Denominators**

Introduction

Rational equations are a fundamental concept in algebra, and they often involve variables in the denominators. In this article, we will explore how to solve rational equations with variable denominators, focusing on the given equation $\frac{1}{x-2} - \frac{5}{x+5} = \frac{7}{x^2+3x-10}$ . We will start by identifying the restrictions on the variable, which are the values that make the denominators zero.

Restrictions on the Variable

To find the restrictions on the variable, we need to determine the values that make the denominators zero. In the given equation, there are three denominators: $x-2$ , $x+5$ , and $x^2+3x-10$ . We will start by factoring the quadratic expression $x^2+3x-10$ .

Factoring the Quadratic Expression

The quadratic expression $x^2+3x-10$ can be factored as follows:

$x^2+3x-10 = (x+5)(x-2)$

Now that we have factored the quadratic expression, we can see that the denominators are $x-2$ , $x+5$ , and $(x+5)(x-2)$ .

Finding the Restrictions

To find the restrictions on the variable, we need to set each denominator equal to zero and solve for $x$ .

For the denominator $x-2$ , we have:

$x-2 = 0$

$x = 2$

For the denominator $x+5$ , we have:

$x+5 = 0$

$x = -5$

For the denominator $(x+5)(x-2)$ , we have:

$(x+5)(x-2) = 0$

$x+5 = 0$ or $x-2 = 0$

$x = -5$ or $x = 2$

Therefore, the restrictions on the variable are $x = -5$ and $x = 2$ .

Solving the Rational Equation

Now that we have identified the restrictions on the variable, we can proceed to solve the rational equation. To do this, we will start by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is $(x-2)(x+5)$ .

Multiplying Both Sides by the LCM

Multiplying both sides of the equation by the LCM $(x-2)(x+5)$ , we get:

$(x-2)(x+5)\left(\frac{1}{x-2} - \frac{5}{x+5}\right) = (x-2)(x+5)\left(\frac{7}{x^2+3x-10}\right)$

Simplifying the equation, we get:

$(x+5) - 5(x-2) = 7$

Expanding and simplifying the equation, we get:

$x+5 - 5x+10 = 7$

$-4x+15 = 7$

$-4x = -8$

$x = 2$

Therefore, the solution to the rational equation is $x = 2$ .

Conclusion

In this article, we have explored how to solve rational equations with variable denominators. We started by identifying the restrictions on the variable, which are the values that make the denominators zero. We then proceeded to solve the rational equation by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. Finally, we simplified the equation and solved for the variable. The solution to the rational equation is $x = 2$ .

Final Answer

Introduction

Rational equations with variable denominators can be challenging to solve, but with the right approach, you can master them. In this article, we will provide a Q&A guide to help you understand and solve rational equations with variable denominators.

Q: What are the restrictions on the variable in a rational equation with variable denominators?

A: The restrictions on the variable are the values that make the denominators zero. To find the restrictions, you need to set each denominator equal to zero and solve for the variable.

Q: How do I find the restrictions on the variable?

A: To find the restrictions on the variable, you need to factor the quadratic expression in the denominator, if possible. Then, set each factor equal to zero and solve for the variable.

Q: What is the least common multiple (LCM) of the denominators?

A: The LCM of the denominators is the product of the factors that make each denominator zero. In other words, it is the product of the factors that make the denominators zero.

Q: How do I multiply both sides of the equation by the LCM?

A: To multiply both sides of the equation by the LCM, you need to multiply each term in the equation by the LCM. This will eliminate the denominators and make it easier to solve the equation.

Q: What are some common mistakes to avoid when solving rational equations with variable denominators?

A: Some common mistakes to avoid when solving rational equations with variable denominators include:

Not identifying the restrictions on the variable
Not factoring the quadratic expression in the denominator
Not multiplying both sides of the equation by the LCM
Not simplifying the equation correctly

Q: How do I simplify the equation after multiplying both sides by the LCM?

A: To simplify the equation after multiplying both sides by the LCM, you need to expand and combine like terms. This will make it easier to solve the equation.

Q: What are some real-world applications of rational equations with variable denominators?

A: Rational equations with variable denominators have many real-world applications, including:

Physics: Rational equations with variable denominators are used to describe the motion of objects in terms of time and distance.
Engineering: Rational equations with variable denominators are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Rational equations with variable denominators are used to model economic systems and make predictions about future trends.

Q: How can I practice solving rational equations with variable denominators?

A: You can practice solving rational equations with variable denominators by working through examples and exercises in your textbook or online resources. You can also try creating your own examples and solving them to test your skills.

Conclusion

In this article, we have provided a Q&A guide to help you understand and solve rational equations with variable denominators. We have covered topics such as restrictions on the variable, finding the LCM, multiplying both sides of the equation by the LCM, and simplifying the equation. We have also discussed common mistakes to avoid and real-world applications of rational equations with variable denominators. By following this guide, you can master rational equations with variable denominators and apply them to real-world problems.

Final Answer

The final answer is that rational equations with variable denominators require careful attention to restrictions, factoring, and simplification. By following the steps outlined in this guide, you can solve these equations and apply them to real-world problems.