Which Expression Is Equivalent To $\frac{3x}{x+1}$ Divided By $x+1$?

Introduction

In algebra, simplifying complex expressions is a crucial skill that helps us solve equations and inequalities more efficiently. One common operation is dividing an algebraic expression by another expression. In this article, we will explore how to simplify the expression $\frac{3x}{x+1}$ divided by $x+1$.

Understanding the Problem

The given expression is $\frac{3x}{x+1}$ divided by $x+1$. To simplify this expression, we need to apply the rules of division and algebraic manipulation. The goal is to simplify the expression into a more manageable form.

Step 1: Apply the Quotient Rule

When dividing an algebraic expression by another expression, we can apply the quotient rule, which states that dividing an expression by a variable is equivalent to multiplying the expression by the reciprocal of the variable. In this case, we can rewrite the expression as:

$$\frac{3x}{x+1} \div (x+1) = \frac{3x}{x+1} \cdot \frac{1}{x+1}$$

Step 2: Simplify the Expression

Now that we have applied the quotient rule, we can simplify the expression by multiplying the numerators and denominators:

$$\frac{3x}{x+1} \cdot \frac{1}{x+1} = \frac{3x \cdot 1}{(x+1) \cdot (x+1)}$$

Step 3: Cancel Out Common Factors

We can simplify the expression further by canceling out common factors in the numerator and denominator:

$$\frac{3x \cdot 1}{(x+1) \cdot (x+1)} = \frac{3x}{(x+1)^2}$$

Conclusion

In conclusion, the expression $\frac{3x}{x+1}$ divided by $x+1$ is equivalent to $\frac{3x}{(x+1)^2}$. This simplified expression is more manageable and easier to work with. By applying the quotient rule and simplifying the expression, we have successfully simplified the given expression.

Real-World Applications

Simplifying complex algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, simplifying expressions can help us model and analyze complex systems, such as the motion of objects or the behavior of electrical circuits. In engineering, simplifying expressions can help us design and optimize systems, such as bridges or electronic circuits. In economics, simplifying expressions can help us model and analyze complex economic systems, such as supply and demand curves.

Tips and Tricks

When simplifying complex algebraic expressions, here are some tips and tricks to keep in mind:

• Apply the quotient rule: When dividing an algebraic expression by another expression, apply the quotient rule to simplify the expression.
• Simplify the expression: Simplify the expression by multiplying the numerators and denominators.
• Cancel out common factors: Cancel out common factors in the numerator and denominator to simplify the expression further.
• Check your work: Check your work by plugging in values or using a calculator to verify the simplified expression.

Common Mistakes to Avoid

When simplifying complex algebraic expressions, here are some common mistakes to avoid:

• Not applying the quotient rule: Failing to apply the quotient rule can lead to incorrect simplifications.
• Not simplifying the expression: Failing to simplify the expression can lead to complex and difficult-to-work-with expressions.
• Not canceling out common factors: Failing to cancel out common factors can lead to unnecessary complexity in the expression.

Conclusion

Introduction

In our previous article, we explored how to simplify the expression $\frac{3x}{x+1}$ divided by $x+1$. In this article, we will answer some frequently asked questions about simplifying complex algebraic expressions.

Q: What is the quotient rule in algebra?

A: The quotient rule in algebra states that dividing an expression by a variable is equivalent to multiplying the expression by the reciprocal of the variable. In other words, dividing an expression by $x$ is the same as multiplying the expression by $\frac{1}{x}$.

Q: How do I apply the quotient rule?

A: To apply the quotient rule, simply multiply the expression by the reciprocal of the variable. For example, if we want to divide $\frac{3x}{x+1}$ by $x+1$, we can rewrite it as $\frac{3x}{x+1} \cdot \frac{1}{x+1}$.

Q: What is the difference between simplifying an expression and canceling out common factors?

A: Simplifying an expression involves combining like terms and eliminating any unnecessary complexity. Canceling out common factors involves eliminating any common factors in the numerator and denominator. While both steps are important, they are distinct and should be performed separately.

Q: How do I know when to simplify an expression and when to cancel out common factors?

A: You should simplify an expression whenever possible to make it easier to work with. You should cancel out common factors whenever you can to eliminate any unnecessary complexity.

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

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

• Not applying the quotient rule
• Not simplifying the expression
• Not canceling out common factors
• Not checking your work

Q: How do I check my work when simplifying complex algebraic expressions?

A: To check your work, plug in values or use a calculator to verify the simplified expression. You can also use algebraic manipulations to verify the expression.

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

A: Simplifying complex algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, simplifying expressions can help us model and analyze complex systems, such as the motion of objects or the behavior of electrical circuits. In engineering, simplifying expressions can help us design and optimize systems, such as bridges or electronic circuits. In economics, simplifying expressions can help us model and analyze complex economic systems, such as supply and demand curves.

Q: How can I practice simplifying complex algebraic expressions?

A: You can practice simplifying complex algebraic expressions by working through examples and exercises. You can also use online resources, such as algebraic calculators or interactive math tools, to help you practice.

Conclusion

In conclusion, simplifying complex algebraic expressions is a crucial skill that helps us solve equations and inequalities more efficiently. By applying the quotient rule, simplifying the expression, and canceling out common factors, we can simplify expressions like $\frac{3x}{x+1}$ divided by $x+1$. Remember to check your work and avoid common mistakes to ensure accurate and efficient simplifications.

Additional Resources

For more information on simplifying complex algebraic expressions, check out the following resources:

• Algebraic calculators: Online calculators that can help you simplify complex algebraic expressions.
• Interactive math tools: Online tools that can help you practice simplifying complex algebraic expressions.
• Algebra textbooks: Textbooks that provide detailed explanations and examples of simplifying complex algebraic expressions.
• Online tutorials: Online tutorials that provide step-by-step instructions on simplifying complex algebraic expressions.