Find The Solution(s) To The Rational Equation $\frac{x-2}{6}=\frac{1}{x+3}$.Select All That Apply.A. $x=-6$B. $x=-4$C. $x=4$D. $x=6$E. $x=-3$F. $x=3$

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Introduction

Rational equations are a type of algebraic equation that involves fractions with variables in the numerator and/or denominator. Solving rational equations can be a challenging task, but with a step-by-step approach, you can find the solution(s) to these equations. In this article, we will focus on solving the rational equation x−26=1x+3\frac{x-2}{6}=\frac{1}{x+3} and provide a detailed explanation of the solution process.

Understanding Rational Equations

A rational equation is an equation that contains one or more fractions with variables in the numerator and/or denominator. Rational equations can be linear or non-linear, and they can involve one or more variables. The key characteristic of rational equations is that they involve fractions, which can make them more challenging to solve.

The Given Rational Equation

The given rational equation is x−26=1x+3\frac{x-2}{6}=\frac{1}{x+3}. This equation involves two fractions with variables in the numerator and/or denominator. Our goal is to find the solution(s) to this equation.

Step 1: Cross-Multiplication

To solve the rational equation, we will start by cross-multiplying the two fractions. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. This will eliminate the fractions and give us a linear equation.

x−26=1x+3\frac{x-2}{6}=\frac{1}{x+3}

Cross-multiplying:

(x−2)(x+3)=6(1)(x-2)(x+3) = 6(1)

Step 2: Expanding and Simplifying

Next, we will expand and simplify the equation by multiplying the terms in the parentheses.

(x−2)(x+3)=6(1)(x-2)(x+3) = 6(1)

Expanding:

x2+3x−2x−6=6x^2 + 3x - 2x - 6 = 6

Simplifying:

x2+x−6=6x^2 + x - 6 = 6

Step 3: Rearranging the Equation

Now, we will rearrange the equation to get all the terms on one side of the equation.

x2+x−6=6x^2 + x - 6 = 6

Subtracting 6 from both sides:

x2+x−12=0x^2 + x - 12 = 0

Step 4: Factoring the Quadratic Equation

The equation x2+x−12=0x^2 + x - 12 = 0 is a quadratic equation that can be factored. Factoring involves expressing the quadratic equation as a product of two binomials.

x2+x−12=0x^2 + x - 12 = 0

Factoring:

(x+4)(x−3)=0(x + 4)(x - 3) = 0

Step 5: Solving for x

Now that we have factored the quadratic equation, we can solve for x by setting each factor equal to zero.

(x+4)(x−3)=0(x + 4)(x - 3) = 0

Setting each factor equal to zero:

x+4=0x + 4 = 0 or x−3=0x - 3 = 0

Solving for x:

x=−4x = -4 or x=3x = 3

Conclusion

In this article, we have solved the rational equation x−26=1x+3\frac{x-2}{6}=\frac{1}{x+3} using a step-by-step approach. We started by cross-multiplying the two fractions, then expanded and simplified the equation, rearranged it to get all the terms on one side, factored the quadratic equation, and finally solved for x. The solution to the equation is x=−4x = -4 or x=3x = 3.

Answer Key

Based on the solution to the equation, the correct answers are:

  • A. x=−6x=-6 is not a solution to the equation.
  • B. x=−4x=-4 is a solution to the equation.
  • C. x=4x=4 is not a solution to the equation.
  • D. x=6x=6 is not a solution to the equation.
  • E. x=−3x=-3 is not a solution to the equation.
  • F. x=3x=3 is a solution to the equation.

Therefore, the correct answers are B and F.

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Introduction

In our previous article, we solved the rational equation x−26=1x+3\frac{x-2}{6}=\frac{1}{x+3} using a step-by-step approach. In this article, we will address some of the most frequently asked questions related to rational equation solutions.

Q&A

Q: What is a rational equation?

A: A rational equation is an equation that contains one or more fractions with variables in the numerator and/or denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, you can follow these steps:

  1. Cross-multiply the two fractions.
  2. Expand and simplify the equation.
  3. Rearrange the equation to get all the terms on one side.
  4. Factor the quadratic equation (if necessary).
  5. Solve for x.

Q: What is cross-multiplication?

A: Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa.

Q: How do I expand and simplify an equation?

A: To expand and simplify an equation, you can multiply the terms in the parentheses and combine like terms.

Q: What is factoring?

A: Factoring involves expressing a quadratic equation as a product of two binomials.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you can look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term.

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

A: A rational equation is an equation that contains one or more fractions with variables in the numerator and/or denominator, while a rational expression is an expression that contains one or more fractions with variables in the numerator and/or denominator.

Q: Can I use a calculator to solve a rational equation?

A: Yes, you can use a calculator to solve a rational equation, but it's always a good idea to check your answer by plugging it back into the original 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 cross-multiplying the fractions
  • Not expanding and simplifying the equation
  • Not rearranging the equation to get all the terms on one side
  • Not factoring the quadratic equation (if necessary)
  • Not solving for x

Conclusion

In this article, we have addressed some of the most frequently asked questions related to rational equation solutions. We hope that this article has provided you with a better understanding of how to solve rational equations and has helped you to avoid some common mistakes.

Additional Resources

If you are looking for additional resources to help you with rational equation solutions, here are a few suggestions:

  • Khan Academy: Rational Equations
  • Mathway: Rational Equations
  • Wolfram Alpha: Rational Equations

Remember, practice makes perfect! The more you practice solving rational equations, the more comfortable you will become with the process.