Solve For { X $} . I F T H E R E I S N O S O L U T I O N , S T A T E T H A T . . If There Is No Solution, State That. . I F T H Ere I S N Oso L U T I O N , S T A T E T Ha T . { \frac{6x}{x+3} - 6 = \frac{2x}{x+3} \}

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Introduction

In this article, we will delve into the world of algebra and solve a complex equation step by step. The given equation is a rational equation, which involves fractions with variables in the numerator and denominator. Our goal is to isolate the variable x and find its value. If there is no solution, we will state that clearly.

The Equation

The given equation is:

6xx+3−6=2xx+3\frac{6x}{x+3} - 6 = \frac{2x}{x+3}

Step 1: Add 6 to Both Sides

To begin solving the equation, we need to get rid of the fraction on the left-hand side. We can do this by adding 6 to both sides of the equation. This will eliminate the fraction and make it easier to work with.

6xx+3−6+6=2xx+3+6\frac{6x}{x+3} - 6 + 6 = \frac{2x}{x+3} + 6

Step 2: Simplify the Left-Hand Side

Now that we have added 6 to both sides, we can simplify the left-hand side of the equation.

6xx+3=2xx+3+6\frac{6x}{x+3} = \frac{2x}{x+3} + 6

Step 3: Subtract 2xx+3\frac{2x}{x+3} from Both Sides

Next, we need to get rid of the fraction on the right-hand side. We can do this by subtracting 2xx+3\frac{2x}{x+3} from both sides of the equation.

6xx+3−2xx+3=2xx+3+6−2xx+3\frac{6x}{x+3} - \frac{2x}{x+3} = \frac{2x}{x+3} + 6 - \frac{2x}{x+3}

Step 4: Simplify the Left-Hand Side

Now that we have subtracted 2xx+3\frac{2x}{x+3} from both sides, we can simplify the left-hand side of the equation.

4xx+3=6\frac{4x}{x+3} = 6

Step 5: Multiply Both Sides by (x+3)(x+3)

To eliminate the fraction, we need to multiply both sides of the equation by (x+3)(x+3).

4x=6(x+3)4x = 6(x+3)

Step 6: Distribute the 6

Next, we need to distribute the 6 to both terms inside the parentheses.

4x=6x+184x = 6x + 18

Step 7: Subtract 4x4x from Both Sides

Now that we have distributed the 6, we can subtract 4x4x from both sides of the equation.

4x−4x=6x−4x+184x - 4x = 6x - 4x + 18

Step 8: Simplify the Equation

Now that we have subtracted 4x4x from both sides, we can simplify the equation.

0=2x+180 = 2x + 18

Step 9: Subtract 18 from Both Sides

Next, we need to get rid of the constant term on the right-hand side. We can do this by subtracting 18 from both sides of the equation.

0−18=2x+18−180 - 18 = 2x + 18 - 18

Step 10: Simplify the Equation

Now that we have subtracted 18 from both sides, we can simplify the equation.

−18=2x-18 = 2x

Step 11: Divide Both Sides by 2

Finally, we need to isolate the variable x. We can do this by dividing both sides of the equation by 2.

−182=2x2\frac{-18}{2} = \frac{2x}{2}

Step 12: Simplify the Equation

Now that we have divided both sides by 2, we can simplify the equation.

−9=x-9 = x

The final answer is −9\boxed{-9}.

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Q: What is the given equation?

A: The given equation is 6xx+3−6=2xx+3\frac{6x}{x+3} - 6 = \frac{2x}{x+3}.

Q: What is the goal of solving the equation?

A: The goal of solving the equation is to isolate the variable x and find its value. If there is no solution, we will state that clearly.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to add 6 to both sides of the equation to eliminate the fraction on the left-hand side.

Q: What is the result of adding 6 to both sides of the equation?

A: The result of adding 6 to both sides of the equation is 6xx+3=2xx+3+6\frac{6x}{x+3} = \frac{2x}{x+3} + 6.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to subtract 2xx+3\frac{2x}{x+3} from both sides of the equation to eliminate the fraction on the right-hand side.

Q: What is the result of subtracting 2xx+3\frac{2x}{x+3} from both sides of the equation?

A: The result of subtracting 2xx+3\frac{2x}{x+3} from both sides of the equation is 4xx+3=6\frac{4x}{x+3} = 6.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to multiply both sides of the equation by (x+3)(x+3) to eliminate the fraction.

Q: What is the result of multiplying both sides of the equation by (x+3)(x+3)?

A: The result of multiplying both sides of the equation by (x+3)(x+3) is 4x=6(x+3)4x = 6(x+3).

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to distribute the 6 to both terms inside the parentheses.

Q: What is the result of distributing the 6?

A: The result of distributing the 6 is 4x=6x+184x = 6x + 18.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to subtract 4x4x from both sides of the equation to isolate the variable x.

Q: What is the result of subtracting 4x4x from both sides of the equation?

A: The result of subtracting 4x4x from both sides of the equation is 0=2x+180 = 2x + 18.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to subtract 18 from both sides of the equation to isolate the variable x.

Q: What is the result of subtracting 18 from both sides of the equation?

A: The result of subtracting 18 from both sides of the equation is −18=2x-18 = 2x.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to divide both sides of the equation by 2 to isolate the variable x.

Q: What is the result of dividing both sides of the equation by 2?

A: The result of dividing both sides of the equation by 2 is −9=x-9 = x.

Q: What is the final answer to the equation?

A: The final answer to the equation is −9\boxed{-9}.

Q: What is the significance of the final answer?

A: The final answer represents the value of the variable x that satisfies the given equation.

Q: Can the equation have multiple solutions?

A: No, the equation has a single solution, which is x = -9.

Q: How can the equation be used in real-world applications?

A: The equation can be used to model real-world situations where the variable x represents a quantity that is related to the given equation.

Q: What are some common mistakes to avoid when solving the equation?

A: Some common mistakes to avoid when solving the equation include:

  • Not following the order of operations
  • Not simplifying the equation at each step
  • Not checking for extraneous solutions

Q: How can the equation be modified to create a new equation?

A: The equation can be modified by changing the coefficients or the variables to create a new equation.

Q: What are some common techniques used to solve equations like this one?

A: Some common techniques used to solve equations like this one include:

  • Adding or subtracting the same value to both sides of the equation
  • Multiplying or dividing both sides of the equation by the same value
  • Using inverse operations to isolate the variable x