Solve The Rational Equation:$\[ \frac{5x}{x+2} = \frac{3x}{x+1} \\]A. \[$ X = -1 \$\]B. \[$ X = -2, 1 \$\]C. \[$ X = 0, \frac{1}{2} \$\]D. \[$ X = \frac{1}{5}, \frac{3}{4} \$\]

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Introduction

Rational equations are a type of algebraic equation that involves fractions with polynomials in both the numerator and denominator. Solving rational equations can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will focus on solving the rational equation 5xx+2=3xx+1\frac{5x}{x+2} = \frac{3x}{x+1}.

Understanding Rational Equations

A rational equation is an equation in which the unknown variable (in this case, xx) appears in the numerator or denominator of a fraction. Rational equations can be solved using various techniques, including factoring, cross-multiplication, and the quadratic formula.

Characteristics of Rational Equations

  • Rational equations involve fractions with polynomials in both the numerator and denominator.
  • The unknown variable (in this case, xx) appears in the numerator or denominator of a fraction.
  • Rational equations can be solved using various techniques, including factoring, cross-multiplication, and the quadratic formula.

Solving the Rational Equation

To solve the rational equation 5xx+2=3xx+1\frac{5x}{x+2} = \frac{3x}{x+1}, we will use the cross-multiplication method.

Step 1: Cross-Multiply

Cross-multiplication involves multiplying both sides of the equation by the denominators of both fractions. This eliminates the fractions and allows us to solve for xx.

5xx+2=3xx+1\frac{5x}{x+2} = \frac{3x}{x+1}

Cross-multiplying:

5x(x+1)=3x(x+2)5x(x+1) = 3x(x+2)

Step 2: Expand and Simplify

Expand and simplify both sides of the equation:

5x2+5x=3x2+6x5x^2 + 5x = 3x^2 + 6x

Step 3: Move All Terms to One Side

Move all terms to one side of the equation:

5x2−3x2+5x−6x=05x^2 - 3x^2 + 5x - 6x = 0

Combine like terms:

2x2−x=02x^2 - x = 0

Step 4: Factor the Quadratic Expression

Factor the quadratic expression:

x(2x−1)=0x(2x - 1) = 0

Step 5: Solve for xx

Solve for xx by setting each factor equal to zero:

x=0x = 0 or 2x−1=02x - 1 = 0

Solving for xx in the second equation:

2x=12x = 1

x=12x = \frac{1}{2}

Conclusion

In conclusion, solving rational equations requires a step-by-step approach. By cross-multiplying, expanding and simplifying, moving all terms to one side, factoring the quadratic expression, and solving for xx, we can find the solutions to the rational equation 5xx+2=3xx+1\frac{5x}{x+2} = \frac{3x}{x+1}. The solutions to this equation are x=0x = 0 and x=12x = \frac{1}{2}.

Final Answer

The final answer is: C\boxed{C}

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Introduction

Solving rational equations can be a challenging task, but with the right approach, it can be broken down into manageable steps. In this article, we will focus on solving rational equations and provide a Q&A guide to help you understand the concepts better.

Q&A: Solving Rational Equations

Q: What is a rational equation?

A: A rational equation is an equation in which the unknown variable (in this case, xx) appears in the numerator or denominator of a fraction.

Q: What are the characteristics of rational equations?

A: Rational equations involve fractions with polynomials in both the numerator and denominator. The unknown variable (in this case, xx) appears in the numerator or denominator of a fraction.

Q: How do I solve a rational equation?

A: To solve a rational equation, you can use various techniques, including factoring, cross-multiplication, and the quadratic formula.

Q: What is cross-multiplication?

A: Cross-multiplication involves multiplying both sides of the equation by the denominators of both fractions. This eliminates the fractions and allows us to solve for xx.

Q: How do I cross-multiply a rational equation?

A: To cross-multiply a rational equation, you multiply both sides of the equation by the denominators of both fractions. For example, if you have the equation 5xx+2=3xx+1\frac{5x}{x+2} = \frac{3x}{x+1}, you would cross-multiply by multiplying both sides by (x+2)(x+1)(x+2)(x+1).

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. The quadratic formula is given by x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: How do I use the quadratic formula to solve a rational equation?

A: To use the quadratic formula to solve a rational equation, you first need to eliminate the fractions by cross-multiplying. Then, you can use the quadratic formula to solve for xx.

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

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

  • Not eliminating the fractions by cross-multiplying
  • Not factoring the quadratic expression
  • Not solving for xx correctly using the quadratic formula

Tips and Tricks

Tip 1: Eliminate the Fractions First

When solving a rational equation, it's essential to eliminate the fractions first by cross-multiplying. This will make it easier to solve for xx.

Tip 2: Factor the Quadratic Expression

Factoring the quadratic expression can help you solve for xx more easily. Look for common factors and try to factor the expression into simpler terms.

Tip 3: Use the Quadratic Formula

If you're stuck solving a quadratic equation, try using the quadratic formula. This will give you two possible solutions for xx.

Conclusion

Solving rational equations requires a step-by-step approach. By understanding the characteristics of rational equations, using cross-multiplication, factoring the quadratic expression, and solving for xx using the quadratic formula, you can find the solutions to rational equations. Remember to avoid common mistakes and use the tips and tricks provided to make solving rational equations easier.

Final Answer

The final answer is: C\boxed{C}