For What Value Of $x$ Is The Rational Expression Below Undefined? 2 X + 4 X − 7 \frac{2x+4}{x-7} X − 7 2 X + 4 A. 2 B. -2 C. 7 D. -7

Mar 9, 2025 by ADMIN 140 views

Solving for the Value of x in a Rational Expression

Understanding Rational Expressions

Rational expressions are a type of algebraic expression that involves the division of two polynomials. They are represented in the form of $\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular expressions. However, rational expressions can be undefined if the denominator is equal to zero.

The Concept of Undefined Rational Expressions

A rational expression is undefined when the denominator is equal to zero. This is because division by zero is undefined in mathematics. In the case of the rational expression $\frac{2x+4}{x-7}$ , we need to find the value of $x$ that makes the denominator equal to zero.

Finding the Value of x

To find the value of $x$ that makes the denominator equal to zero, we need to set the denominator equal to zero and solve for $x$ . In this case, the denominator is $x-7$ , so we set it equal to zero:

$x-7=0$

Solving for x

To solve for $x$ , we need to isolate the variable $x$ on one side of the equation. We can do this by adding 7 to both sides of the equation:

$x-7+7=0+7$

$x=7$

Conclusion

Therefore, the value of $x$ that makes the rational expression $\frac{2x+4}{x-7}$ undefined is $x=7$ . This is because the denominator is equal to zero when $x=7$ , making the rational expression undefined.

Why is this Important?

Understanding when a rational expression is undefined is crucial in algebra and mathematics. It helps us to identify the values of $x$ that make the expression undefined, which can affect the validity of the expression. In this case, we need to be aware that the rational expression $\frac{2x+4}{x-7}$ is undefined when $x=7$ , and we need to take this into account when working with the expression.

Real-World Applications

Rational expressions are used in various real-world applications, such as physics, engineering, and economics. In these fields, rational expressions are used to model real-world phenomena, such as motion, energy, and financial transactions. Understanding when a rational expression is undefined is essential in these applications, as it can affect the accuracy of the model.

Common Mistakes

One common mistake when working with rational expressions is to forget to check if the denominator is equal to zero. This can lead to incorrect solutions and invalid expressions. It is essential to always check the denominator before solving for $x$ .

Tips and Tricks

Here are some tips and tricks to help you solve for $x$ in a rational expression:

Always check the denominator before solving for $x$ .
Set the denominator equal to zero and solve for $x$ .
Use algebraic techniques, such as adding, subtracting, multiplying, and dividing, to isolate the variable $x$ .
Be aware of the values of $x$ that make the expression undefined.

Conclusion

In conclusion, solving for the value of $x$ in a rational expression is a crucial skill in algebra and mathematics. Understanding when a rational expression is undefined is essential in various real-world applications. By following the tips and tricks outlined in this article, you can become proficient in solving for $x$ in rational expressions and avoid common mistakes.

Final Answer

The final answer is $\boxed{7}$ .
Q&A: Solving for x in Rational Expressions

Understanding Rational Expressions

Frequently Asked Questions

Q: What is a rational expression?

A: A rational expression is a type of algebraic expression that involves the division of two polynomials. It is represented in the form of $\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials.

Q: When is a rational expression undefined?

A: A rational expression is undefined when the denominator is equal to zero. This is because division by zero is undefined in mathematics.

Q: How do I find the value of x that makes a rational expression undefined?

A: To find the value of x that makes a rational expression undefined, you need to set the denominator equal to zero and solve for x.

Q: What is the process for solving for x in a rational expression?

A: The process for solving for x in a rational expression involves the following steps:

Set the denominator equal to zero.
Solve for x.
Check if the value of x makes the expression undefined.

Q: What are some common mistakes to avoid when solving for x in a rational expression?

A: Some common mistakes to avoid when solving for x in a rational expression include:

Forgetting to check if the denominator is equal to zero.
Not setting the denominator equal to zero before solving for x.
Not checking if the value of x makes the expression undefined.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to follow these steps:

Factor the numerator and denominator.
Cancel out any common factors.
Simplify the expression.

Q: What are some real-world applications of rational expressions?

A: Rational expressions are used in various real-world applications, such as physics, engineering, and economics. In these fields, rational expressions are used to model real-world phenomena, such as motion, energy, and financial transactions.

Q: How do I determine if a rational expression is undefined?

A: To determine if a rational expression is undefined, you need to check if the denominator is equal to zero. If the denominator is equal to zero, then the expression is undefined.

Q: What is the difference between a rational expression and a fraction?

A: A rational expression is a type of algebraic expression that involves the division of two polynomials, while a fraction is a simple division of two numbers. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular expressions.

Q: How do I add or subtract rational expressions?

A: To add or subtract rational expressions, you need to follow these steps:

Find a common denominator.
Add or subtract the numerators.
Simplify the expression.

Q: What are some tips and tricks for solving for x in rational expressions?

A: Some tips and tricks for solving for x in rational expressions include:

Always check the denominator before solving for x.
Set the denominator equal to zero and solve for x.
Use algebraic techniques, such as adding, subtracting, multiplying, and dividing, to isolate the variable x.
Be aware of the values of x that make the expression undefined.

Conclusion

In conclusion, solving for x in rational expressions is a crucial skill in algebra and mathematics. By understanding when a rational expression is undefined and following the tips and tricks outlined in this article, you can become proficient in solving for x in rational expressions and avoid common mistakes.