Solve The Equation:${ \frac{1}{x+4} + \frac{2}{x+3} = \frac{-1}{x^2 + 7x + 12} }$

Mar 12, 2025 by ADMIN 83 views

**Solving the Equation: A Step-by-Step Guide**

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Introduction

In this article, we will delve into the world of algebra and solve a complex equation involving fractions. The equation we will be solving is $\frac{1}{x+4} + \frac{2}{x+3} = \frac{-1}{x^2 + 7x + 12}$ . This equation requires careful manipulation and simplification to arrive at the solution. We will break down the solution into manageable steps, making it easier to understand and follow along.

Understanding the Equation

The given equation is a rational equation, which means it involves fractions with variables in the denominators. The equation is $\frac{1}{x+4} + \frac{2}{x+3} = \frac{-1}{x^2 + 7x + 12}$ . To solve this equation, we need to first simplify the right-hand side by factoring the denominator.

Factoring the Denominator

The denominator on the right-hand side is $x^2 + 7x + 12$ . We can factor this quadratic expression as $(x+3)(x+4)$ . Therefore, the equation becomes $\frac{1}{x+4} + \frac{2}{x+3} = \frac{-1}{(x+3)(x+4)}$ .

Clearing the Fractions

To clear the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is $(x+3)(x+4)$ . Multiplying both sides by $(x+3)(x+4)$ , we get:

(x+3)(x+4)\left(\frac{1}{x+4} + \frac{2}{x+3}\right) = (x+3)(x+4)\left(\frac{-1}{(x+3)(x+4)}\right)

Simplifying the left-hand side, we get:

(x+3) + 2(x+4) = -1

Simplifying the Equation

Expanding the left-hand side, we get:

x+3 + 2x+8 = -1

Combine like terms:

3x+11 = -1

Isolating the Variable

Subtract 11 from both sides:

3x = -12

Divide both sides by 3:

x = -4

Checking the Solution

To check the solution, we can substitute $x=-4$ back into the original equation. If the equation holds true, then $x=-4$ is the solution.

\frac{1}{-4+4} + \frac{2}{-4+3} = \frac{-1}{(-4)^2 + 7(-4) + 12}

Simplifying the left-hand side, we get:

\frac{1}{0} + \frac{2}{-1} = \frac{-1}{16-28+12}

The left-hand side is undefined, which means $x=-4$ is not a valid solution.

Conclusion

In this article, we solved the equation $\frac{1}{x+4} + \frac{2}{x+3} = \frac{-1}{x^2 + 7x + 12}$ . We broke down the solution into manageable steps, simplifying the equation and isolating the variable. However, upon checking the solution, we found that $x=-4$ is not a valid solution. This highlights the importance of checking the solution to ensure that it satisfies the original equation.

Future Directions

In future articles, we can explore more complex equations and solutions, building on the concepts and techniques learned in this article. We can also delve into the world of calculus, exploring topics such as limits, derivatives, and integrals.

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak

Glossary

Rational Equation: An equation that involves fractions with variables in the denominators.
Least Common Multiple (LCM): The smallest multiple that two or more numbers have in common.
Quadratic Expression: An expression that involves a squared variable, such as $x^2$ .

Note: The above article is a rewritten version of the original content, optimized for readability and SEO. The article includes headings, subheadings, and a glossary to make it easier to understand and navigate.

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Introduction

In our previous article, we solved the equation $\frac{1}{x+4} + \frac{2}{x+3} = \frac{-1}{x^2 + 7x + 12}$ . However, we found that the solution $x=-4$ was not valid. In this article, we will address some common questions and concerns related to solving the equation.

Q&A

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

A: The LCM of the denominators is $(x+3)(x+4)$ .

Q: Why do we need to clear the fractions?

A: We need to clear the fractions to simplify the equation and make it easier to solve.

Q: What is the difference between a rational equation and a quadratic equation?

A: A rational equation is an equation that involves fractions with variables in the denominators, while a quadratic equation is an equation that involves a squared variable.

Q: How do we check the solution?

A: To check the solution, we substitute the value of the variable back into the original equation and see if it holds true.

Q: What if the solution is not valid?

A: If the solution is not valid, it means that the equation does not have a solution or that the solution is extraneous.

Q: Can we use other methods to solve the equation?

A: Yes, we can use other methods such as factoring, the quadratic formula, or graphing to solve the equation.

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

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

Not clearing the fractions
Not checking the solution
Not considering extraneous solutions
Not using the correct method for the type of equation

Conclusion

In this article, we addressed some common questions and concerns related to solving the equation $\frac{1}{x+4} + \frac{2}{x+3} = \frac{-1}{x^2 + 7x + 12}$ . We provided answers to frequently asked questions and highlighted some common mistakes to avoid when solving rational equations.

Future Directions

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Calculus" by Michael Spivak

Glossary

Rational Equation: An equation that involves fractions with variables in the denominators.
Least Common Multiple (LCM): The smallest multiple that two or more numbers have in common.
Quadratic Expression: An expression that involves a squared variable, such as $x^2$ .
Extraneous Solution: A solution that is not valid or does not satisfy the original equation.