What Is The Simplified Expression For The Expression Below?${ -\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28) }$A. { -2x + 3$}$B. { -2x - 3$}$C. { -2x - 8$}$D. { -2x + 8$}$

Understanding the Problem

The given expression is a combination of two fractions, each with a variable term and a constant term. The expression is: ${-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$

To simplify this expression, we need to apply the distributive property and combine like terms.

Step 1: Apply the Distributive Property

The distributive property states that for any real numbers a, b, and c: a(b + c) = ab + ac. We can apply this property to each fraction in the given expression.

For the first fraction, we have: ${-\frac{1}{5}(5x + 20)}$

Using the distributive property, we can rewrite this as: ${-\frac{1}{5}(5x) - \frac{1}{5}(20)}$

Simplifying further, we get: ${-x - 4}$

Similarly, for the second fraction, we have: ${-\frac{1}{4}(4x - 28)}$

Using the distributive property, we can rewrite this as: ${-\frac{1}{4}(4x) + \frac{1}{4}(28)}$

Simplifying further, we get: ${-x + 7}$

Step 2: Combine Like Terms

Now that we have applied the distributive property to each fraction, we can combine like terms.

The expression now becomes: ${-x - 4 - x + 7}$

Combining like terms, we get: ${-2x + 3}$

Conclusion

The simplified expression for the given expression is: ${-2x + 3}$

This matches option A in the multiple-choice question.

Final Answer

The final answer is: $\boxed{-2x + 3}$

Introduction

In the previous section, we simplified the given expression: ${-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$

We applied the distributive property and combined like terms to arrive at the simplified expression: ${-2x + 3}$

In this section, we will address some common questions and concerns related to the solution.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to expand a single term into multiple terms. For example, if we have a term like 3(x + 2), we can use the distributive property to rewrite it as 3x + 6.

Q: Why did we apply the distributive property to each fraction?

A: We applied the distributive property to each fraction to simplify the expression. By expanding each fraction, we were able to combine like terms and arrive at the simplified expression.

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms involves adding or subtracting terms that have the same variable and coefficient. Simplifying an expression, on the other hand, involves rewriting the expression in a more compact or simplified form.

Q: Can you explain the concept of like terms?

A: Like terms are terms that have the same variable and coefficient. For example, 2x and 4x are like terms because they both have the variable x and the same coefficient (2 and 4, respectively).

Q: How do we know which terms to combine when simplifying an expression?

A: When simplifying an expression, we look for terms that have the same variable and coefficient. We then combine these terms by adding or subtracting them.

Q: Can you provide an example of a simplified expression?

A: A simple example of a simplified expression is: 2x + 3 - 2x

When we combine like terms, we get: 3

This is a simplified expression because it is in its most compact form.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is: ${-2x + 3}$

This is the simplified expression for the given expression: ${-\frac{1}{5}(5x + 20) - \frac{1}{4}(4x - 28)}$

Conclusion

In this article, we addressed some common questions and concerns related to the simplified expression. We explained the concept of the distributive property, like terms, and simplifying an expression. We also provided examples and answers to help clarify the solution.

Final Answer

The final answer is: $\boxed{-2x + 3}$