Select The Correct Answer.Which Expression Is Equivalent To The Difference Shown? 5 X + 1 5 X − 4 X + 1 4 X \frac{5x+1}{5x} - \frac{4x+1}{4x} 5 X 5 X + 1 ​ − 4 X 4 X + 1 ​ A. X 20 \frac{x}{20} 20 X ​ B. 1 10 X \frac{1}{10x} 10 X 1 ​ C. − 1 20 X -\frac{1}{20x} − 20 X 1 ​ D. 9 20 X \frac{9}{20x} 20 X 9 ​

Understanding the Problem

When dealing with algebraic expressions, it's essential to simplify them to make calculations easier and more manageable. In this article, we'll focus on simplifying the given expression $\frac{5x+1}{5x} - \frac{4x+1}{4x}$ and selecting the correct equivalent expression from the given options.

Breaking Down the Expression

To simplify the given expression, we need to find a common denominator for the two fractions. The common denominator is the least common multiple (LCM) of the denominators, which in this case is $20x^2$. We can rewrite each fraction with the common denominator:

$$\frac{5x+1}{5x} - \frac{4x+1}{4x} = \frac{(5x+1)(4x)}{20x^2} - \frac{(4x+1)(5x)}{20x^2}$$

Simplifying the Expression

Now that we have a common denominator, we can combine the two fractions by subtracting their numerators:

$$\frac{(5x+1)(4x)}{20x^2} - \frac{(4x+1)(5x)}{20x^2} = \frac{(5x+1)(4x) - (4x+1)(5x)}{20x^2}$$

Expanding the numerators, we get:

$$\frac{(5x+1)(4x) - (4x+1)(5x)}{20x^2} = \frac{20x^2 + 4x - 20x^2 - 5x}{20x^2}$$

Simplifying the numerator, we get:

$$\frac{20x^2 + 4x - 20x^2 - 5x}{20x^2} = \frac{-x}{20x^2}$$

Further Simplification

We can simplify the expression further by factoring out a negative sign from the numerator:

$$\frac{-x}{20x^2} = -\frac{x}{20x^2}$$

We can also simplify the denominator by canceling out a common factor of $x$:

$$-\frac{x}{20x^2} = -\frac{1}{20x}$$

Selecting the Correct Answer

Now that we have simplified the expression, we can compare it to the given options to select the correct answer. The simplified expression is $-\frac{1}{20x}$, which matches option C.

Conclusion

In this article, we simplified the given expression $\frac{5x+1}{5x} - \frac{4x+1}{4x}$ and selected the correct equivalent expression from the given options. We used a step-by-step approach to simplify the expression, including finding a common denominator, combining the fractions, and simplifying the numerator. The final simplified expression is $-\frac{1}{20x}$, which matches option C.

Key Takeaways

• To simplify an algebraic expression, find a common denominator for the fractions.
• Combine the fractions by subtracting their numerators.
• Simplify the numerator by combining like terms.
• Factor out any common factors from the numerator and denominator.
• Cancel out any common factors from the numerator and denominator.

Practice Problems

• Simplify the expression $\frac{2x+3}{2x} - \frac{x+2}{x}$.
• Simplify the expression $\frac{3x-2}{3x} - \frac{2x-3}{2x}$.
• Simplify the expression $\frac{x+2}{x} - \frac{2x+3}{2x}$.

Answer Key

• The simplified expression is $\frac{1}{2x}$.
• The simplified expression is $\frac{1}{6x}$.
• The simplified expression is $-\frac{1}{2x}$.

Additional Resources

• For more practice problems and solutions, visit our website at [insert website URL].
• For additional resources and study guides, visit our online store at [insert online store URL].
  

Understanding the Basics

Algebraic expressions can be complex and intimidating, but with the right tools and techniques, they can be simplified and made more manageable. In this article, we'll answer some common questions about simplifying algebraic expressions and provide tips and tricks for tackling even the most challenging problems.

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

A: The first step in simplifying an algebraic expression is to find a common denominator for the fractions. This will allow you to combine the fractions and simplify the expression.

Q: How do I find a common denominator?

A: To find a common denominator, you need to find the least common multiple (LCM) of the denominators. This can be done by listing the multiples of each denominator and finding the smallest number that appears in both lists.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, the LCM of 2 and 3 is 6, because 6 is the smallest number that is a multiple of both 2 and 3.

Q: How do I combine fractions with different denominators?

A: To combine fractions with different denominators, you need to find a common denominator and then add or subtract the numerators. For example, if you have the fractions 1/2 and 1/3, you can find a common denominator of 6 and then add the numerators: 3/6 + 2/6 = 5/6.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves reducing it to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true. For example, simplifying the expression 2x + 3 involves combining like terms and reducing it to its simplest form, while solving the equation 2x + 3 = 5 involves finding the value of x that makes the equation true.

Q: How do I know when an expression is simplified?

A: An expression is simplified when it cannot be reduced further. This means that there are no like terms that can be combined, and the expression is in its simplest form.

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

A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Forgetting to find a common denominator when combining fractions
• Not combining like terms
• Not simplifying the expression to its simplest form
• Making errors when multiplying or dividing fractions

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through practice problems and exercises. You can also try simplifying expressions on your own and then checking your work to make sure you got the correct answer.

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

A: Simplifying algebraic expressions has many real-world applications, including:

• Calculating the area and perimeter of shapes
• Finding the volume of solids
• Solving problems in physics and engineering
• Working with financial data and statistics

Conclusion

Simplifying algebraic expressions is an essential skill for anyone who wants to work with math and science. By understanding the basics of simplifying expressions and practicing regularly, you can become more confident and proficient in your ability to simplify even the most complex expressions.

Additional Resources

• For more practice problems and solutions, visit our website at [insert website URL].
• For additional resources and study guides, visit our online store at [insert online store URL].
• For help with specific math problems or questions, visit our online forum at [insert online forum URL].