5. What Is The Difference Of $ \frac{15x}{2x-4} - \frac{3}{2x-4} $?A. $ \frac{45x}{2x-4}, \, X \neq 2 $B. $ \frac{15x-3}{(2x-4)^2}, \, X \neq 2 $C. $ \frac{15x-3}{2x-4}, \, X \neq 2 $D. $ \frac{12x}{2x-4}, \, X
Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the difference of two fractions. We will use the given problem as a case study to demonstrate the step-by-step process of simplifying algebraic expressions.
The Problem
The problem asks us to find the difference of two fractions:
Step 1: Factor Out the Common Denominator
The first step in simplifying the expression is to factor out the common denominator, which is . We can rewrite the expression as:
Step 2: Simplify the Numerator
The next step is to simplify the numerator by combining like terms. In this case, we can combine the terms and to get:
Step 3: Check for Common Factors
Before we can simplify the expression further, we need to check if there are any common factors in the numerator and denominator. In this case, there are no common factors, so we can proceed to the next step.
Step 4: Simplify the Expression
The final step is to simplify the expression by canceling out any common factors. In this case, there are no common factors, so the expression is already simplified.
The Final Answer
Therefore, the difference of the two fractions is:
Conclusion
Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the step-by-step process outlined in this article, we can simplify even the most complex expressions. In this case, we used the given problem to demonstrate the process of simplifying the difference of two fractions.
Answer Key
The correct answer is:
- C. $ \frac{15x-3}{2x-4}, , x \neq 2 $
Additional Tips and Resources
- To simplify algebraic expressions, it's essential to factor out the common denominator and simplify the numerator.
- Check for common factors in the numerator and denominator before simplifying the expression.
- Use the step-by-step process outlined in this article to simplify even the most complex expressions.
- For more practice problems and resources, check out the following websites:
- Khan Academy: Algebra
- Mathway: Algebra
- IXL: Algebra
Final Thoughts
Q: What is the first step in simplifying an algebraic expression?
A: The first step in simplifying an algebraic expression is to factor out the common denominator. This involves identifying the common factor in the numerator and denominator and rewriting the expression with the common factor factored out.
Q: How do I simplify the numerator of an algebraic expression?
A: To simplify the numerator of an algebraic expression, you need to combine like terms. This involves adding or subtracting the coefficients of the same variables. For example, if you have the expression , you can combine the like terms to get .
Q: What is the difference between a common factor and a common denominator?
A: A common factor is a factor that is present in both the numerator and denominator of an algebraic expression. A common denominator, on the other hand, is the denominator that is present in both fractions of an algebraic expression.
Q: How do I check for common factors in an algebraic expression?
A: To check for common factors in an algebraic expression, you need to look for factors that are present in both the numerator and denominator. You can do this by factoring the numerator and denominator and looking for common factors.
Q: What is the final step in simplifying an algebraic expression?
A: The final step in simplifying an algebraic expression is to simplify the expression by canceling out any common factors. This involves dividing both the numerator and denominator by the common factor.
Q: Can I simplify an algebraic expression if there are no common factors?
A: Yes, you can simplify an algebraic expression even if there are no common factors. In this case, the expression is already simplified, and you can leave it as is.
Q: How do I know if an algebraic expression is simplified?
A: An algebraic expression is simplified if there are no common factors in the numerator and denominator. You can check this by factoring the numerator and denominator and looking for common factors.
Q: What are some common mistakes to avoid when simplifying algebraic expressions?
A: Some common mistakes to avoid when simplifying algebraic expressions include:
- Not factoring out the common denominator
- Not combining like terms in the numerator
- Not checking for common factors
- Not simplifying the expression by canceling out common factors
Q: How can I practice simplifying algebraic expressions?
A: You can practice simplifying algebraic expressions by working through practice problems and exercises. You can find these online or in algebra textbooks. You can also try simplifying expressions on your own and checking your work with a calculator or a friend.
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
- Determining the cost of goods and services
- Solving problems in physics and engineering
- Analyzing data and making predictions
Q: Can I use a calculator to simplify algebraic expressions?
A: Yes, you can use a calculator to simplify algebraic expressions. However, it's also important to understand the underlying math and be able to simplify expressions by hand. This will help you to develop a deeper understanding of the math and to identify any errors that may have been made.