Simplify The Expression: 4 X + 1 4 X + 6 X − 11 4 X \frac{4x+1}{4x}+\frac{6x-11}{4x} 4 X 4 X + 1 ​ + 4 X 6 X − 11 ​

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When dealing with fractions, combining them can be a bit challenging, but with the right approach, it can be done easily. In this article, we will simplify the expression $\frac{4x+1}{4x}+\frac{6x-11}{4x}$ using basic algebraic operations.

Understanding the Expression

The given expression consists of two fractions with the same denominator, which is $4x$. To simplify this expression, we need to combine the two fractions by adding their numerators while keeping the common denominator.

Combining the Fractions

To combine the fractions, we need to follow the order of operations (PEMDAS):

1. Parentheses: None in this case.
2. Exponents: None in this case.
3. Multiplication and Division: Divide the numerator and denominator of each fraction by their greatest common divisor (GCD).
4. Addition and Subtraction: Add the numerators of the two fractions.

Let's simplify the expression step by step:

Step 1: Find the GCD of the Numerators

The GCD of $4x+1$ and $6x-11$ is $1$, since they have no common factors other than $1$.

Step 2: Divide the Numerators by the GCD

Divide the numerators by the GCD:

$\frac{4x+1}{1} = 4x+1$

$\frac{6x-11}{1} = 6x-11$

Step 3: Add the Numerators

Add the numerators of the two fractions:

$(4x+1) + (6x-11) = 10x-10$

Step 4: Write the Simplified Expression

Now that we have added the numerators, we can write the simplified expression:

$\frac{4x+1}{4x}+\frac{6x-11}{4x} = \frac{10x-10}{4x}$

Simplifying the Expression Further

We can simplify the expression further by dividing the numerator and denominator by their GCD, which is $2$.

$\frac{10x-10}{4x} = \frac{5x-5}{2x}$

Conclusion

In this article, we simplified the expression $\frac{4x+1}{4x}+\frac{6x-11}{4x}$ using basic algebraic operations. We combined the two fractions by adding their numerators while keeping the common denominator, and then simplified the expression further by dividing the numerator and denominator by their GCD. The final simplified expression is $\frac{5x-5}{2x}$.

Final Answer

The final answer is $\boxed{\frac{5x-5}{2x}}$.

Related Topics

• Simplifying expressions with fractions
• Combining fractions with the same denominator
• Algebraic operations with fractions

Example Problems

• Simplify the expression $\frac{3x+2}{3x}+\frac{5x-1}{3x}$
• Simplify the expression $\frac{2x-3}{2x}+\frac{4x+1}{2x}$

Practice Problems

• Simplify the expression $\frac{6x+1}{6x}+\frac{8x-2}{6x}$
• Simplify the expression $\frac{3x-2}{3x}+\frac{5x+1}{3x}$
  

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Introduction

In our previous article, we simplified the expression $\frac{4x+1}{4x}+\frac{6x-11}{4x}$ using basic algebraic operations. In this article, we will answer some frequently asked questions related to simplifying expressions with fractions.

Q&A

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

A: The first step in simplifying an expression with fractions is to identify the common denominator. If the fractions have the same denominator, you can combine them by adding their numerators.

Q: How do I find the common denominator of two fractions?

A: To find the common denominator of two fractions, you need to find the least common multiple (LCM) of the two denominators. If the denominators are the same, the LCM is the denominator itself.

Q: What is the order of operations for simplifying expressions with fractions?

A: The order of operations for simplifying expressions with fractions is:

1. Parentheses: None in this case.
2. Exponents: None in this case.
3. Multiplication and Division: Divide the numerator and denominator of each fraction by their greatest common divisor (GCD).
4. Addition and Subtraction: Add the numerators of the two fractions.

Q: How do I simplify an expression with fractions that have different denominators?

A: To simplify an expression with fractions that have different denominators, you need to find the least common multiple (LCM) of the two denominators and rewrite each fraction with the LCM as the denominator.

Q: What is the final simplified expression for $\frac{4x+1}{4x}+\frac{6x-11}{4x}$?

A: The final simplified expression for $\frac{4x+1}{4x}+\frac{6x-11}{4x}$ is $\frac{5x-5}{2x}$.

Q: Can I simplify an expression with fractions that have variables in the denominator?

A: Yes, you can simplify an expression with fractions that have variables in the denominator. However, you need to be careful when dividing the numerator and denominator by their greatest common divisor (GCD).

Q: How do I check if my simplified expression is correct?

A: To check if your simplified expression is correct, you can plug in a value for the variable and simplify the expression. If the result is the same as the original expression, then your simplified expression is correct.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with fractions. We covered topics such as finding the common denominator, simplifying expressions with different denominators, and checking the correctness of a simplified expression.

Final Answer

The final answer is $\boxed{\frac{5x-5}{2x}}$.

Related Topics

• Simplifying expressions with fractions
• Combining fractions with the same denominator
• Algebraic operations with fractions

Example Problems

• Simplify the expression $\frac{3x+2}{3x}+\frac{5x-1}{3x}$
• Simplify the expression $\frac{2x-3}{2x}+\frac{4x+1}{2x}$

Practice Problems

• Simplify the expression $\frac{6x+1}{6x}+\frac{8x-2}{6x}$
• Simplify the expression $\frac{3x-2}{3x}+\frac{5x+1}{3x}$