Simplify:${ \frac{16x - 1}{(x - 1)(x + 4)} - \frac{14x - 4}{(x - 1)(x + 4)} }${ \square\$}

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Introduction

When dealing with algebraic expressions, simplification is a crucial step in solving equations and inequalities. In this article, we will focus on simplifying a given expression involving fractions. The expression to be simplified is $\frac{16x - 1}{(x - 1)(x + 4)} - \frac{14x - 4}{(x - 1)(x + 4)}$. We will use various techniques to simplify this expression and provide a final result.

Understanding the Expression

The given expression involves two fractions with the same denominator, $(x - 1)(x + 4)$. To simplify this expression, we can start by finding a common denominator for both fractions. However, since the denominators are the same, we can directly subtract the numerators.

Subtracting the Numerators

To subtract the numerators, we need to have the same denominator for both fractions. Since the denominators are the same, we can directly subtract the numerators.

$\frac{16x - 1}{(x - 1)(x + 4)} - \frac{14x - 4}{(x - 1)(x + 4)} = \frac{(16x - 1) - (14x - 4)}{(x - 1)(x + 4)}$

Simplifying the Numerator

Now, we can simplify the numerator by combining like terms.

$\frac{(16x - 1) - (14x - 4)}{(x - 1)(x + 4)} = \frac{16x - 1 - 14x + 4}{(x - 1)(x + 4)}$

Combining Like Terms

We can combine like terms in the numerator by adding or subtracting the coefficients of the same variables.

$\frac{16x - 1 - 14x + 4}{(x - 1)(x + 4)} = \frac{2x + 3}{(x - 1)(x + 4)}$

Final Result

The simplified expression is $\frac{2x + 3}{(x - 1)(x + 4)}$. This is the final result after simplifying the given expression.

Conclusion

In this article, we simplified a given expression involving fractions. We used various techniques such as subtracting the numerators and combining like terms to simplify the expression. The final result is $\frac{2x + 3}{(x - 1)(x + 4)}$. This expression can be used as a starting point for further algebraic manipulations or as a solution to a specific problem.

Tips and Tricks

• When dealing with fractions, it's essential to have a common denominator to add or subtract the numerators.
• Combining like terms is a crucial step in simplifying algebraic expressions.
• Always check the final result to ensure that it's in the simplest form possible.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications in various fields such as physics, engineering, and economics. For example, in physics, simplifying expressions is crucial in solving problems involving motion, energy, and momentum. In engineering, simplifying expressions is essential in designing and analyzing complex systems. In economics, simplifying expressions is necessary in modeling and analyzing economic systems.

Common Mistakes

• Failing to have a common denominator when adding or subtracting fractions.
• Not combining like terms in the numerator.
• Not checking the final result to ensure that it's in the simplest form possible.

Final Thoughts

Simplifying algebraic expressions is a crucial step in solving equations and inequalities. By using various techniques such as subtracting the numerators and combining like terms, we can simplify complex expressions and arrive at a final result. This article provided a step-by-step guide on simplifying a given expression involving fractions. We hope that this article has provided valuable insights and techniques for simplifying algebraic expressions.

Additional Resources

For further learning and practice, we recommend the following resources:

• Khan Academy: Algebra
• MIT OpenCourseWare: Algebra
• Wolfram Alpha: Algebra

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for further learning and practice. They are not directly related to the content of this article.

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Introduction

In our previous article, we simplified a given expression involving fractions. We used various techniques such as subtracting the numerators and combining like terms to simplify the expression. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q&A

Q: What is the first step in simplifying an expression involving fractions?

A: The first step in simplifying an expression involving fractions is to find a common denominator. However, if the denominators are the same, we can directly subtract the numerators.

Q: How do I combine like terms in the numerator?

A: To combine like terms in the numerator, we need to add or subtract the coefficients of the same variables. For example, if we have $2x + 3x$, we can combine them to get $5x$.

Q: What is the final result of the expression $\frac{16x - 1}{(x - 1)(x + 4)} - \frac{14x - 4}{(x - 1)(x + 4)}$?

A: The final result of the expression is $\frac{2x + 3}{(x - 1)(x + 4)}$.

Q: Can I use this technique to simplify other expressions involving fractions?

A: Yes, this technique can be used to simplify other expressions involving fractions. However, you need to make sure that the denominators are the same before subtracting the numerators.

Q: What are some common mistakes to avoid when simplifying expressions involving fractions?

A: Some common mistakes to avoid when simplifying expressions involving fractions include failing to have a common denominator, not combining like terms in the numerator, and not checking the final result to ensure that it's in the simplest form possible.

Q: How can I practice simplifying expressions involving fractions?

A: You can practice simplifying expressions involving fractions by using online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha. You can also try solving problems from algebra textbooks or online resources.

Q: What are some real-world applications of simplifying expressions involving fractions?

A: Simplifying expressions involving fractions has numerous real-world applications in various fields such as physics, engineering, and economics. For example, in physics, simplifying expressions is crucial in solving problems involving motion, energy, and momentum. In engineering, simplifying expressions is essential in designing and analyzing complex systems. In economics, simplifying expressions is necessary in modeling and analyzing economic systems.

Conclusion

In this article, we provided a Q&A section to address any questions or concerns that readers may have. We hope that this article has provided valuable insights and techniques for simplifying expressions involving fractions. Remember to practice regularly and use online resources to improve your skills.

Tips and Tricks

• Always check the final result to ensure that it's in the simplest form possible.
• Use online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha to practice simplifying expressions involving fractions.
• Try solving problems from algebra textbooks or online resources to improve your skills.

Real-World Applications

Simplifying expressions involving fractions has numerous real-world applications in various fields such as physics, engineering, and economics. For example, in physics, simplifying expressions is crucial in solving problems involving motion, energy, and momentum. In engineering, simplifying expressions is essential in designing and analyzing complex systems. In economics, simplifying expressions is necessary in modeling and analyzing economic systems.

Common Mistakes

• Failing to have a common denominator when adding or subtracting fractions.
• Not combining like terms in the numerator.
• Not checking the final result to ensure that it's in the simplest form possible.

Final Thoughts

Simplifying expressions involving fractions is a crucial step in solving equations and inequalities. By using various techniques such as subtracting the numerators and combining like terms, we can simplify complex expressions and arrive at a final result. This article provided a Q&A section to address any questions or concerns that readers may have. We hope that this article has provided valuable insights and techniques for simplifying expressions involving fractions.

Additional Resources

For further learning and practice, we recommend the following resources:

• Khan Academy: Algebra
• MIT OpenCourseWare: Algebra
• Wolfram Alpha: Algebra

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for further learning and practice. They are not directly related to the content of this article.