For Which Value(s) Of $x$ Will The Rational Expression Below Be Undefined? Check All That Apply. ( X − 3 ) ( X + 6 ) X + 7 \frac{(x-3)(x+6)}{x+7} X + 7 ( X − 3 ) ( X + 6 ) ​ A. 3 B. -7 C. -3 D. -6 E. 6 F. 7

A rational expression is undefined when its denominator equals zero. In the given expression $\frac{(x-3)(x+6)}{x+7}$, we need to find the values of $x$ that make the denominator $x+7$ equal to zero.

Finding the Values of $x$

To find the values of $x$ that make the denominator $x+7$ equal to zero, we set the denominator equal to zero and solve for $x$.

$$x+7=0$$

Subtracting 7 from both sides gives us:

$$x=-7$$

Therefore, the value of $x$ that makes the denominator $x+7$ equal to zero is $-7$.

Checking the Options

Now, let's check the options to see which ones match the value we found.

• A. 3: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.
• B. -7: This value makes the denominator $x+7$ equal to zero, so it is the correct answer.
• C. -3: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.
• D. -6: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.
• E. 6: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.
• F. 7: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.

Conclusion

In conclusion, the value of $x$ that makes the rational expression $\frac{(x-3)(x+6)}{x+7}$ undefined is $-7$.

Additional Considerations

It's worth noting that the numerator of the rational expression is $(x-3)(x+6)$. This means that the rational expression will also be undefined if the numerator equals zero. However, we are only asked to find the values of $x$ that make the denominator equal to zero.

Final Answer

The final answer is $\boxed{-7}$.

Other Possible Values of $x$

In addition to the value $-7$, there are other values of $x$ that make the rational expression undefined. These values are the roots of the numerator $(x-3)(x+6)$.

Finding the Roots of the Numerator

To find the roots of the numerator, we set the numerator equal to zero and solve for $x$.

$$(x-3)(x+6)=0$$

This equation is true when either $x-3=0$ or $x+6=0$.

Solving for $x$

Solving for $x$ in the first equation gives us:

$$x-3=0$$

Adding 3 to both sides gives us:

$$x=3$$

Solving for $x$ in the second equation gives us:

$$x+6=0$$

Subtracting 6 from both sides gives us:

$$x=-6$$

Therefore, the values of $x$ that make the numerator $(x-3)(x+6)$ equal to zero are $3$ and $-6$.

Checking the Options

Now, let's check the options to see which ones match the values we found.

• A. 3: This value makes the numerator $(x-3)(x+6)$ equal to zero, so it is the correct answer.
• B. -7: This value makes the denominator $x+7$ equal to zero, so it is the correct answer.
• C. -3: This value does not make the numerator $(x-3)(x+6)$ equal to zero, so it is not the correct answer.
• D. -6: This value makes the numerator $(x-3)(x+6)$ equal to zero, so it is the correct answer.
• E. 6: This value does not make the numerator $(x-3)(x+6)$ equal to zero, so it is not the correct answer.
• F. 7: This value does not make the numerator $(x-3)(x+6)$ equal to zero, so it is not the correct answer.

Conclusion

In conclusion, the values of $x$ that make the rational expression $\frac{(x-3)(x+6)}{x+7}$ undefined are $-7$, $3$, and $-6$.

Final Answer

The final answer is $\boxed{-7, 3, -6}$.

Other Possible Values of $x$

In addition to the values $-7$, $3$, and $-6$, there are other values of $x$ that make the rational expression undefined. These values are the roots of the denominator $x+7$.

Finding the Roots of the Denominator

To find the roots of the denominator, we set the denominator equal to zero and solve for $x$.

$$x+7=0$$

Subtracting 7 from both sides gives us:

$$x=-7$$

Therefore, the value of $x$ that makes the denominator $x+7$ equal to zero is $-7$.

Checking the Options

Now, let's check the options to see which ones match the value we found.

• A. 3: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.
• B. -7: This value makes the denominator $x+7$ equal to zero, so it is the correct answer.
• C. -3: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.
• D. -6: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.
• E. 6: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.
• F. 7: This value does not make the denominator $x+7$ equal to zero, so it is not the correct answer.

Conclusion

In conclusion, the value of $x$ that makes the rational expression $\frac{(x-3)(x+6)}{x+7}$ undefined is $-7$.

Final Answer

The final answer is $\boxed{-7}$.

Other Possible Values of $x$

In addition to the value $-7$, there are other values of $x$ that make the rational expression undefined. These values are the roots of the numerator $(x-3)(x+6)$.

Finding the Roots of the Numerator

To find the roots of the numerator, we set the numerator equal to zero and solve for $x$.

$$(x-3)(x+6)=0$$

This equation is true when either $x-3=0$ or $x+6=0$.

Solving for $x$

Solving for $x$ in the first equation gives us:

$$x-3=0$$

Adding 3 to both sides gives us:

$$x=3$$

Solving for $x$ in the second equation gives us:

$$x+6=0$$

Subtracting 6 from both sides gives us:

$$x=-6$$

Therefore, the values of $x$ that make the numerator $(x-3)(x+6)$ equal to zero are $3$ and $-6$.

Checking the Options

Now, let's check the options to see which ones match the values we found.

• A. 3: This value makes the numerator $(x-3)(x+6)$ equal to zero, so it is the correct answer.
• B. -7: This value makes the denominator $x+7$ equal to zero, so it is the correct answer.
• C. -3: This value does not make the numerator $(x-3)(x+6)$ equal to zero, so it is not the correct answer.
• D. -6: This value makes the numerator $(x-3)(x+6)$ equal to zero, so it is the correct answer.
• E. 6: This value does not make the numerator $(x-3)(x+6)$ equal to zero, so it is not the correct answer.
• F. 7: This value does not make the numerator $(x-3)(x+6)$ equal to zero, so it is not the correct answer.

Conclusion

In conclusion, the values of $x$ that make the rational expression $\frac{(x-3)(x+6)}{x+7}$ undefined are $-7$, $3$, and $-6$.

Final Answer

The final answer is $\boxed{-7, 3, -6}$.

Other Possible Values of $x$

In addition to the values $-7$, $3$, and $-6$, there are other values of $x$ that make the rational expression undefined. These values are the roots of the denominator $x+7$.

Finding the Roots of the Denominator

To find the roots of the denominator, we set the denominator equal to zero and solve for $x$.

x<br/> **Q&A: Rational Expression Undefined Values** =============================================

Q: What is a rational expression?

A: A rational expression is a mathematical expression that is the ratio of two polynomials.

Q: What makes a rational expression undefined?

A: A rational expression is undefined when its denominator equals zero.

Q: How do you find the values of $x$ that make a rational expression undefined?

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

Q: What is the denominator of the rational expression $\frac{(x-3)(x+6)}{x+7}$?

A: The denominator of the rational expression $\frac{(x-3)(x+6)}{x+7}$ is $x+7$.

Q: What value of $x$ makes the denominator $x+7$ equal to zero?

A: The value of $x$ that makes the denominator $x+7$ equal to zero is $-7$.

Q: What values of $x$ make the numerator $(x-3)(x+6)$ equal to zero?

A: The values of $x$ that make the numerator $(x-3)(x+6)$ equal to zero are $3$ and $-6$.

Q: What values of $x$ make the rational expression $\frac{(x-3)(x+6)}{x+7}$ undefined?

A: The values of $x$ that make the rational expression $\frac{(x-3)(x+6)}{x+7}$ undefined are $-7$, $3$, and $-6$.

Q: How do you check the options to see which ones match the values you found?

A: To check the options, you need to plug in each value into the rational expression and see if it equals zero.

Q: What is the final answer to the problem?

A: The final answer is $\boxed{-7, 3, -6}$.

Additional Q&A

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

A: A rational number is a number that can be expressed as the ratio of two integers, while a rational expression is a mathematical expression that is the ratio of two polynomials.

Q: Can a rational expression have more than one denominator?

A: Yes, a rational expression can have more than one denominator.

Q: How do you simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, and then cancel out any common factors.

Q: What is the importance of finding the values of $x$ that make a rational expression undefined?

A: Finding the values of $x$ that make a rational expression undefined is important because it helps you to identify the values of $x$ that are not in the domain of the rational expression.

Conclusion

In conclusion, finding the values of $x$ that make a rational expression undefined is an important step in solving rational expression problems. By following the steps outlined in this article, you can find the values of $x$ that make a rational expression undefined and simplify the rational expression.

Final Answer

The final answer is $\boxed{-7, 3, -6}$.