Simplify The Rational Expression:${ \frac{5}{x+1} - \frac{8}{x-7} }$

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will focus on simplifying the rational expression 5x+1−8x−7\frac{5}{x+1} - \frac{8}{x-7}. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding Rational Expressions

A rational expression is a fraction that contains variables in the numerator or denominator. Rational expressions can be simplified by factoring the numerator and denominator, canceling out common factors, and combining the remaining terms.

Characteristics of Rational Expressions

  • A rational expression is a fraction that contains variables in the numerator or denominator.
  • Rational expressions can be simplified by factoring the numerator and denominator.
  • Common factors can be canceled out to simplify the expression.
  • The remaining terms can be combined to simplify the expression further.

Simplifying the Rational Expression

To simplify the rational expression 5x+1−8x−7\frac{5}{x+1} - \frac{8}{x-7}, we need to follow these steps:

Step 1: Find the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest multiple of the denominators of the two fractions. In this case, the denominators are x+1x+1 and x−7x-7. The LCD is (x+1)(x−7)(x+1)(x-7).

Step 2: Rewrite the Fractions with the LCD

We need to rewrite the fractions with the LCD as the denominator. This can be done by multiplying the numerator and denominator of each fraction by the necessary factors.

5x+1=5(x−7)(x+1)(x−7)\frac{5}{x+1} = \frac{5(x-7)}{(x+1)(x-7)}

8x−7=8(x+1)(x+1)(x−7)\frac{8}{x-7} = \frac{8(x+1)}{(x+1)(x-7)}

Step 3: Subtract the Fractions

Now that the fractions have the same denominator, we can subtract them.

5(x−7)(x+1)(x−7)−8(x+1)(x+1)(x−7)\frac{5(x-7)}{(x+1)(x-7)} - \frac{8(x+1)}{(x+1)(x-7)}

=5(x−7)−8(x+1)(x+1)(x−7)= \frac{5(x-7) - 8(x+1)}{(x+1)(x-7)}

Step 4: Simplify the Expression

We can simplify the expression by combining like terms in the numerator.

=5x−35−8x−8(x+1)(x−7)= \frac{5x - 35 - 8x - 8}{(x+1)(x-7)}

=−3x−43(x+1)(x−7)= \frac{-3x - 43}{(x+1)(x-7)}

Conclusion

Simplifying rational expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, we can simplify the rational expression 5x+1−8x−7\frac{5}{x+1} - \frac{8}{x-7}. The final simplified expression is −3x−43(x+1)(x−7)\frac{-3x - 43}{(x+1)(x-7)}.

Example Use Cases

Rational expressions are used in a variety of real-world applications, including:

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

Tips and Tricks

Here are some tips and tricks for simplifying rational expressions:

  • Factor the numerator and denominator: Factoring the numerator and denominator can help you identify common factors that can be canceled out.
  • Use the distributive property: The distributive property can be used to expand the numerator and denominator, making it easier to identify common factors.
  • Combine like terms: Combining like terms in the numerator can help simplify the expression.

Common Mistakes

Here are some common mistakes to avoid when simplifying rational expressions:

  • Not factoring the numerator and denominator: Failing to factor the numerator and denominator can make it difficult to identify common factors.
  • Not using the distributive property: Failing to use the distributive property can make it difficult to expand the numerator and denominator.
  • Not combining like terms: Failing to combine like terms in the numerator can make the expression more complicated than it needs to be.

Final Thoughts

Simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article, we can simplify the rational expression 5x+1−8x−7\frac{5}{x+1} - \frac{8}{x-7}. The final simplified expression is −3x−43(x+1)(x−7)\frac{-3x - 43}{(x+1)(x-7)}. With practice and patience, you can become proficient in simplifying rational expressions and apply this skill to a variety of real-world applications.

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Frequently Asked Questions

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables in the numerator or denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

  1. Find the least common denominator (LCD) of the fractions.
  2. Rewrite the fractions with the LCD as the denominator.
  3. Subtract the fractions.
  4. Simplify the expression by combining like terms in the numerator.

Q: What is the least common denominator (LCD)?

A: The least common denominator (LCD) is the smallest multiple of the denominators of the two fractions.

Q: How do I find the LCD?

A: To find the LCD, you need to list the multiples of each denominator and find the smallest multiple that is common to both lists.

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

A: A rational number is a fraction that has a finite decimal representation, whereas a rational expression is a fraction that contains variables in the numerator or denominator.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator by following the same steps as before.

Q: How do I handle a rational expression with a negative exponent?

A: To handle a rational expression with a negative exponent, you need to rewrite the expression with a positive exponent by taking the reciprocal of the fraction.

Q: Can I simplify a rational expression with a fraction in the numerator?

A: Yes, you can simplify a rational expression with a fraction in the numerator by following the same steps as before.

Q: How do I handle a rational expression with a variable in the numerator and denominator?

A: To handle a rational expression with a variable in the numerator and denominator, you need to follow the same steps as before.

Q: Can I simplify a rational expression with a complex fraction?

A: Yes, you can simplify a rational expression with a complex fraction by following the same steps as before.

Example Questions

Q: Simplify the rational expression 3x+2−2x−3\frac{3}{x+2} - \frac{2}{x-3}.

A: To simplify the rational expression, we need to follow the steps outlined above.

  1. Find the least common denominator (LCD) of the fractions.
  2. Rewrite the fractions with the LCD as the denominator.
  3. Subtract the fractions.
  4. Simplify the expression by combining like terms in the numerator.

The final simplified expression is 3x−6−2x+6(x+2)(x−3)=x(x+2)(x−3)\frac{3x - 6 - 2x + 6}{(x+2)(x-3)} = \frac{x}{(x+2)(x-3)}.

Q: Simplify the rational expression 2xx+1−3x−2\frac{2x}{x+1} - \frac{3}{x-2}.

A: To simplify the rational expression, we need to follow the steps outlined above.

  1. Find the least common denominator (LCD) of the fractions.
  2. Rewrite the fractions with the LCD as the denominator.
  3. Subtract the fractions.
  4. Simplify the expression by combining like terms in the numerator.

The final simplified expression is 2x(x−2)−3(x+1)(x+1)(x−2)=2x2−4x−3x−3(x+1)(x−2)=2x2−7x−3(x+1)(x−2)\frac{2x(x-2) - 3(x+1)}{(x+1)(x-2)} = \frac{2x^2 - 4x - 3x - 3}{(x+1)(x-2)} = \frac{2x^2 - 7x - 3}{(x+1)(x-2)}.

Tips and Tricks

Here are some tips and tricks for simplifying rational expressions:

  • Factor the numerator and denominator: Factoring the numerator and denominator can help you identify common factors that can be canceled out.
  • Use the distributive property: The distributive property can be used to expand the numerator and denominator, making it easier to identify common factors.
  • Combine like terms: Combining like terms in the numerator can help simplify the expression.
  • Check for common factors: Check for common factors in the numerator and denominator to see if they can be canceled out.

Common Mistakes

Here are some common mistakes to avoid when simplifying rational expressions:

  • Not factoring the numerator and denominator: Failing to factor the numerator and denominator can make it difficult to identify common factors.
  • Not using the distributive property: Failing to use the distributive property can make it difficult to expand the numerator and denominator.
  • Not combining like terms: Failing to combine like terms in the numerator can make the expression more complicated than it needs to be.
  • Not checking for common factors: Failing to check for common factors in the numerator and denominator can make it difficult to simplify the expression.

Final Thoughts

Simplifying rational expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article and practicing with example questions, you can become proficient in simplifying rational expressions and apply this skill to a variety of real-world applications. Remember to factor the numerator and denominator, use the distributive property, combine like terms, and check for common factors to simplify rational expressions.