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

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Introduction

When dealing with rational expressions, it's essential to understand that the expression can equal zero only when the numerator is equal to zero, and the denominator is not equal to zero. This is because a fraction is equal to zero only when its numerator is zero, and its denominator is not zero. In this article, we will explore the values of $x$ for which the rational expression $\frac{(x-3)(x+6)}{x+7}$ will equal zero.

Setting the Numerator Equal to Zero

To find the values of $x$ for which the rational expression will equal zero, we need to set the numerator equal to zero and solve for $x$. The numerator of the rational expression is $(x-3)(x+6)$. Setting this expression equal to zero gives us the equation:

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

Solving the Equation

To solve the equation $(x-3)(x+6) = 0$, we need to find the values of $x$ that make each factor equal to zero. This gives us two separate equations:

$$x-3 = 0$$

and

$$x+6 = 0$$

Solving the First Equation

Solving the first equation $x-3 = 0$ gives us:

$$x = 3$$

Solving the Second Equation

Solving the second equation $x+6 = 0$ gives us:

$$x = -6$$

Checking the Denominator

Before we can conclude that the values of $x$ we found are the solutions to the original equation, we need to check that the denominator is not equal to zero for these values of $x$. The denominator of the rational expression is $x+7$. We need to check that $x+7 \neq 0$ for $x = 3$ and $x = -6$.

Checking the Denominator for $x = 3$

Substituting $x = 3$ into the denominator $x+7$ gives us:

$$3+7 = 10$$

Since $10 \neq 0$, we can conclude that $x = 3$ is a solution to the original equation.

Checking the Denominator for $x = -6$

Substituting $x = -6$ into the denominator $x+7$ gives us:

$$-6+7 = 1$$

Since $1 \neq 0$, we can conclude that $x = -6$ is a solution to the original equation.

Conclusion

In conclusion, the values of $x$ for which the rational expression $\frac{(x-3)(x+6)}{x+7}$ will equal zero are $x = 3$ and $x = -6$. These values make the numerator equal to zero, and the denominator is not equal to zero.

Final Answer

The final answer is:

• A. 6: Incorrect, because $x = 6$ is not a solution to the original equation.
• B. -3: Incorrect, because $x = -3$ is not a solution to the original equation.
• C. 7: Incorrect, because $x = 7$ is not a solution to the original equation.
• D. -6: Correct, because $x = -6$ is a solution to the original equation.
• E. 3: Correct, because $x = 3$ is a solution to the original equation.
• F. -7: Incorrect, because $x = -7$ is not a solution to the original equation.

Additional Discussion

It's worth noting that the rational expression $\frac{(x-3)(x+6)}{x+7}$ will never equal zero when $x = -7$, because the denominator will be equal to zero. This is an important consideration when working with rational expressions, as it can affect the validity of the solutions we find.

Final Thoughts

In conclusion, the values of $x$ for which the rational expression $\frac{(x-3)(x+6)}{x+7}$ will equal zero are $x = 3$ and $x = -6$. These values make the numerator equal to zero, and the denominator is not equal to zero. It's essential to consider the denominator when working with rational expressions, as it can affect the validity of the solutions we find.

Introduction

In the previous article, we explored the values of $x$ for which the rational expression $\frac{(x-3)(x+6)}{x+7}$ will equal zero. In this article, we will answer some frequently asked questions about rational expressions and provide additional insights into the topic.

Q&A

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be used to represent a wide range of mathematical concepts, including algebraic functions and equations.

Q: How do I 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. For example, the rational expression $\frac{(x-3)(x+6)}{x+7}$ can be simplified by factoring the numerator and denominator, and then canceling out the common factor $(x+7)$.

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, such as $\frac{1}{2}$ or $\frac{3}{4}$. A rational expression, on the other hand, is a fraction that contains variables and/or constants in the numerator and/or denominator.

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

A: A rational expression is undefined when the denominator is equal to zero. To determine if a rational expression is undefined, you need to check if the denominator is equal to zero for any value of the variable.

Q: Can a rational expression have multiple solutions?

A: Yes, a rational expression can have multiple solutions. For example, the rational expression $\frac{(x-3)(x+6)}{x+7}$ has two solutions: $x = 3$ and $x = -6$.

Q: How do I graph a rational expression?

A: To graph a rational expression, you need to identify the x-intercepts, y-intercepts, and any vertical asymptotes. You can use a graphing calculator or software to graph the rational expression.

Q: Can a rational expression be used to model real-world problems?

A: Yes, rational expressions can be used to model real-world problems. For example, the rational expression $\frac{(x-3)(x+6)}{x+7}$ can be used to model the height of a projectile as a function of time.

Additional Discussion

Rational expressions are a fundamental concept in algebra and are used to model a wide range of real-world problems. By understanding how to simplify, graph, and solve rational expressions, you can gain a deeper understanding of mathematical concepts and develop problem-solving skills.

Final Thoughts

In conclusion, rational expressions are a powerful tool for modeling and solving mathematical problems. By understanding the basics of rational expressions, you can gain a deeper understanding of mathematical concepts and develop problem-solving skills. Whether you're a student or a professional, rational expressions are an essential part of mathematics and can be used to model a wide range of real-world problems.

Common Mistakes to Avoid

When working with rational expressions, it's essential to avoid common mistakes such as:

• Not factoring the numerator and denominator
• Not canceling out common factors
• Not checking if the denominator is equal to zero
• Not identifying the x-intercepts, y-intercepts, and vertical asymptotes

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Final Answer

The final answer is:

• A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.
• A rational expression can be simplified by factoring the numerator and denominator and canceling out common factors.
• A rational expression is undefined when the denominator is equal to zero.
• A rational expression can have multiple solutions.
• A rational expression can be used to model real-world problems.

Additional Resources

For more information on rational expressions, check out the following resources:

• Khan Academy: Rational Expressions
• Mathway: Rational Expressions
• Wolfram Alpha: Rational Expressions

By using these resources, you can gain a deeper understanding of rational expressions and develop problem-solving skills.