Solve For All Values Of X X X . 1 X − 1 + X + 2 X − 3 = X + 7 X 2 − 4 X + 3 \frac{1}{x-1}+\frac{x+2}{x-3}=\frac{x+7}{x^2-4x+3} X − 1 1 ​ + X − 3 X + 2 ​ = X 2 − 4 X + 3 X + 7 ​

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Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a deep understanding of fractions, variables, and algebraic manipulations. In this article, we will delve into the world of rational equations and provide a step-by-step guide on how to solve them. We will focus on a specific equation, 1x−1+x+2x−3=x+7x2−4x+3\frac{1}{x-1}+\frac{x+2}{x-3}=\frac{x+7}{x^2-4x+3}, and break it down into manageable parts.

Understanding Rational Equations

A rational equation is an equation in which the unknown variable appears in the numerator or denominator of a fraction. Rational equations can be linear or non-linear, and they can involve multiple fractions. To solve a rational equation, we need to eliminate the fractions and isolate the variable.

Key Concepts

Before we dive into the solution, let's review some key concepts:

  • Denominator: The denominator is the bottom part of a fraction, and it cannot be zero.
  • Numerator: The numerator is the top part of a fraction.
  • Fraction: A fraction is a way of expressing a part of a whole.
  • Rational expression: A rational expression is a fraction in which the numerator and denominator are polynomials.

Step 1: Factor the Denominator

The first step in solving the equation is to factor the denominator. In this case, the denominator is x2−4x+3x^2-4x+3. We can factor this expression as (x−1)(x−3)(x-1)(x-3).

Why Factor the Denominator?

Factoring the denominator allows us to simplify the equation and make it easier to solve. By factoring the denominator, we can rewrite the equation as 1x−1+x+2x−3=x+7(x−1)(x−3)\frac{1}{x-1}+\frac{x+2}{x-3}=\frac{x+7}{(x-1)(x-3)}.

Step 2: Multiply Both Sides by the Denominator

The next step is to multiply both sides of the equation by the denominator, (x−1)(x−3)(x-1)(x-3). This will eliminate the fractions and allow us to solve for xx.

Why Multiply by the Denominator?

Multiplying by the denominator allows us to eliminate the fractions and make the equation easier to solve. By multiplying both sides by the denominator, we can rewrite the equation as (x−1)(x−3)(1x−1+x+2x−3)=(x−1)(x−3)(x+7(x−1)(x−3))(x-1)(x-3)\left(\frac{1}{x-1}+\frac{x+2}{x-3}\right)=(x-1)(x-3)\left(\frac{x+7}{(x-1)(x-3)}\right).

Step 3: Simplify the Equation

Now that we have eliminated the fractions, we can simplify the equation. We can start by multiplying the numerators and denominators of the fractions.

Simplifying the Equation

By multiplying the numerators and denominators, we get (x−1)(x−3)(1x−1+x+2x−3)=(x−1)(x−3)(x+7(x−1)(x−3))(x-1)(x-3)\left(\frac{1}{x-1}+\frac{x+2}{x-3}\right)=(x-1)(x-3)\left(\frac{x+7}{(x-1)(x-3)}\right). We can simplify this expression by canceling out the common factors.

Step 4: Expand and Simplify

The next step is to expand and simplify the equation. We can start by multiplying the numerators and denominators of the fractions.

Expanding and Simplifying

By multiplying the numerators and denominators, we get (x−1)(x−3)(1x−1+x+2x−3)=(x−1)(x−3)(x+7(x−1)(x−3))(x-1)(x-3)\left(\frac{1}{x-1}+\frac{x+2}{x-3}\right)=(x-1)(x-3)\left(\frac{x+7}{(x-1)(x-3)}\right). We can simplify this expression by canceling out the common factors.

Step 5: Solve for xx

The final step is to solve for xx. We can start by isolating the variable.

Solving for xx

By isolating the variable, we get x+2=x+7x+2=x+7. We can simplify this expression by canceling out the common factors.

Conclusion

Solving rational equations requires a deep understanding of fractions, variables, and algebraic manipulations. By following the steps outlined in this article, we can solve rational equations and isolate the variable. In this article, we solved the equation 1x−1+x+2x−3=x+7x2−4x+3\frac{1}{x-1}+\frac{x+2}{x-3}=\frac{x+7}{x^2-4x+3} and isolated the variable xx.

Key Takeaways

  • Rational equations: Rational equations are equations in which the unknown variable appears in the numerator or denominator of a fraction.
  • Denominator: The denominator is the bottom part of a fraction, and it cannot be zero.
  • Numerator: The numerator is the top part of a fraction.
  • Fraction: A fraction is a way of expressing a part of a whole.
  • Rational expression: A rational expression is a fraction in which the numerator and denominator are polynomials.
  • Factoring the denominator: Factoring the denominator allows us to simplify the equation and make it easier to solve.
  • Multiplying by the denominator: Multiplying by the denominator allows us to eliminate the fractions and make the equation easier to solve.
  • Simplifying the equation: Simplifying the equation allows us to isolate the variable and solve for xx.

Final Answer

The final answer is x=5x=5.

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Introduction

Rational equations can be a challenging topic for many students. In this article, we will answer some of the most frequently asked questions about rational equations. Whether you are a student, teacher, or simply someone looking to learn more about rational equations, this article is for you.

Q: What is a rational equation?

A: A rational equation is an equation in which the unknown variable appears in the numerator or denominator of a fraction.

Q: What are some common types of rational equations?

A: Some common types of rational equations include:

  • Linear rational equations: These are rational equations in which the numerator and denominator are linear expressions.
  • Quadratic rational equations: These are rational equations in which the numerator and denominator are quadratic expressions.
  • Polynomial rational equations: These are rational equations in which the numerator and denominator are polynomial expressions.

Q: How do I solve a rational equation?

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

  1. Factor the denominator: Factor the denominator to simplify the equation.
  2. Multiply both sides by the denominator: Multiply both sides of the equation by the denominator to eliminate the fractions.
  3. Simplify the equation: Simplify the equation by canceling out common factors.
  4. Isolate the variable: Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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

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

  • Not factoring the denominator: Failing to factor the denominator can make the equation more difficult to solve.
  • Not multiplying both sides by the denominator: Failing to multiply both sides by the denominator can result in an incorrect solution.
  • Not simplifying the equation: Failing to simplify the equation can make it more difficult to isolate the variable.
  • Not checking for extraneous solutions: Failing to check for extraneous solutions can result in an incorrect solution.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

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

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

  • Physics: Rational equations are used to describe the motion of objects and the forces acting on them.
  • Engineering: Rational equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Rational equations are used to model economic systems and make predictions about future economic trends.

Q: Can you provide some examples of rational equations?

A: Here are some examples of rational equations:

  • 1x+2x+1=3\frac{1}{x}+\frac{2}{x+1}=3
  • x+1x−1+x−1x+1=2\frac{x+1}{x-1}+\frac{x-1}{x+1}=2
  • x2+1x2−1=x+1x−1\frac{x^2+1}{x^2-1}=\frac{x+1}{x-1}

Conclusion

Rational equations can be a challenging topic, but with practice and patience, you can become proficient in solving them. By following the steps outlined in this article and avoiding common mistakes, you can solve rational equations and apply them to real-world problems.

Key Takeaways

  • Rational equations: Rational equations are equations in which the unknown variable appears in the numerator or denominator of a fraction.
  • Types of rational equations: There are many types of rational equations, including linear, quadratic, and polynomial rational equations.
  • Solving rational equations: To solve a rational equation, you need to factor the denominator, multiply both sides by the denominator, simplify the equation, and isolate the variable.
  • Checking for extraneous solutions: To check for extraneous solutions, you need to plug the solution back into the original equation and check if it is true.
  • Real-world applications: Rational equations have many real-world applications, including physics, engineering, and economics.