Which Inequality Represents All Values Of $x$ For Which The Quotient Below Is Defined? 15 ( X − 1 ) 2 X 2 \frac{\sqrt{15(x-1)}}{\sqrt{2x^2}} 2 X 2 15 ( X − 1 ) A. X ≤ − 1 X \leq -1 X ≤ − 1 B. X \textgreater 1 X \ \textgreater \ 1 X \textgreater 1 C. X ≥ 1 X \geq 1 X ≥ 1 D. $x \ \textless

Mar 13, 2025 by ADMIN 300 views

**Understanding the Conditions for a Defined Quotient**

When dealing with mathematical expressions, it's essential to consider the conditions under which the expression is defined. In this case, we're given a quotient involving square roots, and we need to determine the inequality that represents all values of $x$ for which the quotient is defined.

The Quotient and Its Conditions

The given quotient is $\frac{\sqrt{15(x-1)}}{\sqrt{2x^2}}$ . For this quotient to be defined, both the numerator and the denominator must be non-negative, and the denominator cannot be equal to zero.

Numerator Conditions

The numerator is $\sqrt{15(x-1)}$ . Since the square root of a number is non-negative, we need to ensure that $15(x-1) \geq 0$ . This inequality can be rewritten as $x-1 \geq 0$ , which simplifies to $x \geq 1$ .

Denominator Conditions

The denominator is $\sqrt{2x^2}$ . Since the square root of a number is non-negative, we need to ensure that $2x^2 \geq 0$ . This inequality is always true for all real values of $x$ , since the square of any real number is non-negative.

Denominator Cannot Be Zero

However, we also need to ensure that the denominator is not equal to zero. This means that $\sqrt{2x^2} \neq 0$ . Since the square root of a number is non-zero only if the number itself is non-zero, we need to ensure that $2x^2 \neq 0$ . This inequality is equivalent to $x \neq 0$ .

Combining the Conditions

Now that we have considered the conditions for the numerator and the denominator, we can combine them to determine the inequality that represents all values of $x$ for which the quotient is defined. We need to ensure that $x \geq 1$ and $x \neq 0$ . Since $x \neq 0$ is implied by $x \geq 1$ , we can simplify the inequality to $x \geq 1$ .

Conclusion

In conclusion, the inequality that represents all values of $x$ for which the quotient $\frac{\sqrt{15(x-1)}}{\sqrt{2x^2}}$ is defined is $x \geq 1$ .

Answer

The correct answer is C. $x \geq 1$ .

Additional Considerations

It's worth noting that the inequality $x \geq 1$ is not the only possible solution. We also need to consider the case where $x < 0$ . In this case, the numerator $\sqrt{15(x-1)}$ is not defined, since the square root of a negative number is not a real number. Therefore, the inequality $x \geq 1$ is the only possible solution.

Graphical Representation

To visualize the solution, we can graph the inequality $x \geq 1$ on a number line. The graph will consist of a single point at $x = 1$ , and all points to the right of this point.

Real-World Applications

The concept of defining a quotient and determining the conditions under which it is defined has many real-world applications. For example, in physics, the quotient of two quantities may represent a ratio of forces or energies. In engineering, the quotient of two quantities may represent a ratio of resistances or impedances. In economics, the quotient of two quantities may represent a ratio of prices or costs.

Conclusion

In our previous article, we explored the conditions under which the quotient $\frac{\sqrt{15(x-1)}}{\sqrt{2x^2}}$ is defined. Now, let's dive deeper into the world of quotients and answer some frequently asked questions.

Q: What is the main condition for a quotient to be defined?

A: The main condition for a quotient to be defined is that the denominator cannot be equal to zero. Additionally, both the numerator and the denominator must be non-negative.

Q: What happens if the numerator is negative?

A: If the numerator is negative, the square root of the numerator will not be a real number, and the quotient will not be defined.

Q: What happens if the denominator is zero?

A: If the denominator is zero, the quotient will be undefined, since division by zero is not allowed.

Q: Can a quotient have a negative denominator?

A: Yes, a quotient can have a negative denominator, but the quotient will still be defined as long as the numerator is non-negative.

Q: What is the difference between a quotient and a fraction?

A: A quotient and a fraction are essentially the same thing. A quotient is the result of dividing one quantity by another, while a fraction is a way of representing a quotient as a ratio of two numbers.

Q: Can a quotient have a variable in the denominator?

A: Yes, a quotient can have a variable in the denominator, but the variable must be non-zero for the quotient to be defined.

Q: How do I simplify a quotient?

A: To simplify a quotient, you can start by factoring the numerator and the denominator, and then cancel out any common factors.

Q: Can a quotient be equal to zero?

A: Yes, a quotient can be equal to zero if the numerator is zero and the denominator is non-zero.

Q: What is the difference between a quotient and a product?

A: A quotient and a product are two different mathematical operations. A quotient is the result of dividing one quantity by another, while a product is the result of multiplying two or more quantities together.

Q: Can a quotient have a negative numerator and a negative denominator?

A: Yes, a quotient can have a negative numerator and a negative denominator, but the quotient will still be defined as long as the numerator and the denominator are not both zero.

Conclusion

In conclusion, quotients are an essential part of mathematics, and understanding the conditions under which they are defined is crucial for solving problems and simplifying expressions. By answering these frequently asked questions, we hope to have provided a better understanding of quotients and their applications.

Real-World Applications

Quotients have many real-world applications, including:

Physics: Quotients are used to represent ratios of forces or energies.
Engineering: Quotients are used to represent ratios of resistances or impedances.
Economics: Quotients are used to represent ratios of prices or costs.
Finance: Quotients are used to represent ratios of returns or risks.

Conclusion

In conclusion, quotients are a fundamental concept in mathematics, and understanding their conditions and applications is essential for solving problems and making informed decisions. By exploring the world of quotients, we hope to have provided a better understanding of this important mathematical concept.