Simplify The Expression:${ \frac{2x - 9}{x - 5} \cdot \frac{4x - 20}{x + 6} }$

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying the given expression: $\frac{2x - 9}{x - 5} \cdot \frac{4x - 20}{x + 6}$. We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex algebraic expressions.

Understanding the Expression

Before we dive into the simplification process, let's take a closer look at the given expression. We have two fractions multiplied together: $\frac{2x - 9}{x - 5}$ and $\frac{4x - 20}{x + 6}$. Our goal is to simplify this expression by combining the two fractions and eliminating any common factors.

Step 1: Factorize the Numerators and Denominators

To simplify the expression, we need to factorize the numerators and denominators of both fractions. Let's start with the first fraction: $\frac{2x - 9}{x - 5}$. We can factorize the numerator as $2(x - 4.5)$ and the denominator as $(x - 5)$. Similarly, for the second fraction: $\frac{4x - 20}{x + 6}$, we can factorize the numerator as $4(x - 5)$ and the denominator as $(x + 6)$.

Step 2: Cancel Out Common Factors

Now that we have factorized the numerators and denominators, we can cancel out any common factors between the two fractions. Looking at the first fraction, we have $2(x - 4.5)$ in the numerator and $(x - 5)$ in the denominator. We can cancel out the $(x - 5)$ term, leaving us with $2$ in the numerator. Similarly, for the second fraction, we have $4(x - 5)$ in the numerator and $(x + 6)$ in the denominator. We can cancel out the $(x - 5)$ term, leaving us with $4$ in the numerator.

Step 3: Multiply the Remaining Factors

Now that we have canceled out the common factors, we are left with two fractions: $\frac{2}{1}$ and $\frac{4}{x + 6}$. To simplify the expression, we need to multiply these two fractions together. Multiplying the numerators gives us $2 \cdot 4 = 8$, and multiplying the denominators gives us $1 \cdot (x + 6) = x + 6$.

Step 4: Simplify the Final Expression

Now that we have multiplied the remaining factors, we are left with the final expression: $\frac{8}{x + 6}$. This is the simplified form of the original expression.

Conclusion

Simplifying algebraic expressions is an essential skill that requires patience, practice, and persistence. By breaking down the problem into manageable steps and using factorization and cancellation techniques, we can simplify even the most complex expressions. In this article, we have walked through the step-by-step process of simplifying the expression: $\frac{2x - 9}{x - 5} \cdot \frac{4x - 20}{x + 6}$. We hope that this article has provided you with a clear understanding of how to simplify complex algebraic expressions and has given you the confidence to tackle more challenging problems in the future.

Frequently Asked Questions

• What is algebraic manipulation? Algebraic manipulation is the process of simplifying or transforming algebraic expressions using various techniques such as factorization, cancellation, and multiplication.
• Why is simplifying algebraic expressions important? Simplifying algebraic expressions is important because it helps to make complex problems more manageable and easier to solve. It also helps to identify patterns and relationships between variables, which can be useful in a variety of applications.
• What are some common techniques used in algebraic manipulation? Some common techniques used in algebraic manipulation include factorization, cancellation, and multiplication. These techniques can be used to simplify expressions, eliminate common factors, and solve equations.

Real-World Applications

Algebraic manipulation has a wide range of real-world applications, including:

• Science and Engineering: Algebraic manipulation is used to model and solve complex problems in physics, engineering, and other scientific fields.
• Economics: Algebraic manipulation is used to model and analyze economic systems, including supply and demand curves and economic growth models.
• Computer Science: Algebraic manipulation is used in computer science to solve complex problems in algorithms, data structures, and software engineering.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Use factorization: Factorization is a powerful technique for simplifying expressions. Look for common factors in the numerator and denominator, and cancel them out.
• Cancel out common factors: Canceling out common factors is a key step in simplifying expressions. Look for common factors in the numerator and denominator, and cancel them out.
• Multiply the remaining factors: Once you have canceled out common factors, multiply the remaining factors together to simplify the expression.

Conclusion

Simplifying algebraic expressions is an essential skill that requires patience, practice, and persistence. By breaking down the problem into manageable steps and using factorization and cancellation techniques, we can simplify even the most complex expressions. In this article, we have walked through the step-by-step process of simplifying the expression: $\frac{2x - 9}{x - 5} \cdot \frac{4x - 20}{x + 6}$. We hope that this article has provided you with a clear understanding of how to simplify complex algebraic expressions and has given you the confidence to tackle more challenging problems in the future.

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and professional should possess. In this article, we will focus on simplifying the given expression: $\frac{2x - 9}{x - 5} \cdot \frac{4x - 20}{x + 6}$. We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex algebraic expressions.

Q&A: Simplifying Algebraic Expressions

Q: What is algebraic manipulation?

A: Algebraic manipulation is the process of simplifying or transforming algebraic expressions using various techniques such as factorization, cancellation, and multiplication.

Q: Why is simplifying algebraic expressions important?

A: Simplifying algebraic expressions is important because it helps to make complex problems more manageable and easier to solve. It also helps to identify patterns and relationships between variables, which can be useful in a variety of applications.

Q: What are some common techniques used in algebraic manipulation?

A: Some common techniques used in algebraic manipulation include factorization, cancellation, and multiplication. These techniques can be used to simplify expressions, eliminate common factors, and solve equations.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to follow these steps:

1. Factorize the numerator and denominator of the expression.
2. Cancel out any common factors between the numerator and denominator.
3. Multiply the remaining factors together to simplify the expression.

Q: What is factorization?

A: Factorization is the process of breaking down an algebraic expression into its simplest form by identifying common factors. For example, the expression $2x^2 + 4x$ can be factorized as $2x(x + 2)$.

Q: What is cancellation?

A: Cancellation is the process of eliminating common factors between the numerator and denominator of an algebraic expression. For example, the expression $\frac{2x}{x}$ can be simplified by canceling out the common factor $x$, leaving us with $2$.

Q: How do I multiply algebraic expressions?

A: To multiply algebraic expressions, you need to follow these steps:

1. Multiply the numerators together.
2. Multiply the denominators together.
3. Simplify the resulting expression by canceling out any 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 canceling out common factors between the numerator and denominator.
• Not multiplying the numerators and denominators together correctly.
• Not simplifying the resulting expression by canceling out any common factors.

Real-World Applications

Algebraic manipulation has a wide range of real-world applications, including:

• Science and Engineering: Algebraic manipulation is used to model and solve complex problems in physics, engineering, and other scientific fields.
• Economics: Algebraic manipulation is used to model and analyze economic systems, including supply and demand curves and economic growth models.
• Computer Science: Algebraic manipulation is used in computer science to solve complex problems in algorithms, data structures, and software engineering.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Use factorization: Factorization is a powerful technique for simplifying expressions. Look for common factors in the numerator and denominator, and cancel them out.
• Cancel out common factors: Canceling out common factors is a key step in simplifying expressions. Look for common factors in the numerator and denominator, and cancel them out.
• Multiply the remaining factors: Once you have canceled out common factors, multiply the remaining factors together to simplify the expression.

Conclusion

Simplifying algebraic expressions is an essential skill that requires patience, practice, and persistence. By breaking down the problem into manageable steps and using factorization and cancellation techniques, we can simplify even the most complex expressions. In this article, we have walked through the step-by-step process of simplifying the expression: $\frac{2x - 9}{x - 5} \cdot \frac{4x - 20}{x + 6}$. We hope that this article has provided you with a clear understanding of how to simplify complex algebraic expressions and has given you the confidence to tackle more challenging problems in the future.

Frequently Asked Questions

• What is algebraic manipulation? Algebraic manipulation is the process of simplifying or transforming algebraic expressions using various techniques such as factorization, cancellation, and multiplication.
• Why is simplifying algebraic expressions important? Simplifying algebraic expressions is important because it helps to make complex problems more manageable and easier to solve. It also helps to identify patterns and relationships between variables, which can be useful in a variety of applications.
• What are some common techniques used in algebraic manipulation? Some common techniques used in algebraic manipulation include factorization, cancellation, and multiplication. These techniques can be used to simplify expressions, eliminate common factors, and solve equations.

Real-World Applications

Algebraic manipulation has a wide range of real-world applications, including:

• Science and Engineering: Algebraic manipulation is used to model and solve complex problems in physics, engineering, and other scientific fields.
• Economics: Algebraic manipulation is used to model and analyze economic systems, including supply and demand curves and economic growth models.
• Computer Science: Algebraic manipulation is used in computer science to solve complex problems in algorithms, data structures, and software engineering.

Tips and Tricks

Here are some tips and tricks to help you simplify algebraic expressions:

• Use factorization: Factorization is a powerful technique for simplifying expressions. Look for common factors in the numerator and denominator, and cancel them out.
• Cancel out common factors: Canceling out common factors is a key step in simplifying expressions. Look for common factors in the numerator and denominator, and cancel them out.
• Multiply the remaining factors: Once you have canceled out common factors, multiply the remaining factors together to simplify the expression.