The Quotient and Its Conditions

Oct 13, 2025 by leolavanderia.com.br 122 views

Which Inequality Represents All Values of x for Which the Quotient is Defined?

Understanding the Problem

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 $\sqrt{6 x^2} \div \sqrt{4 x}$ . For this quotient to be defined, the expressions inside the square roots must be non-negative, and the denominator cannot be equal to zero.

Non-Negativity of Expressions Inside Square Roots

The expression inside the first square root, $6 x^2$ , is always non-negative for any real value of x, since it's a squared term. The expression inside the second square root, $4 x$ , is non-negative if and only if $x \geq 0$ .

Denominator Cannot Be Equal to Zero

The denominator of the quotient is $\sqrt{4 x}$ . For the quotient to be defined, the denominator cannot be equal to zero. This means that $4 x \neq 0$ , which implies that $x \neq 0$ .

Combining Conditions

Combining the conditions for non-negativity and the condition that the denominator cannot be equal to zero, we get that the quotient is defined if and only if $x \geq 0$ and $x \neq 0$ . Since $x \neq 0$ is implied by $x \geq 0$ , the final condition is simply $x \geq 0$ .

Conclusion

The inequality that represents all values of x for which the quotient $\sqrt{6 x^2} \div \sqrt{4 x}$ is defined is $x \geq 0$ .

Answer

The correct answer is C. $x \geq 0$ .

Additional Insights

It's worth noting that the inequality $x \geq 0$ is the most restrictive condition that ensures the quotient is defined. This is because the expression inside the first square root is always non-negative, and the denominator cannot be equal to zero. Therefore, the quotient is defined for all non-negative values of x.

Comparison with Other Options

Let's compare the correct answer with the other options:

Option A: $x \ \textgreater \ 1$ is not correct, since the quotient is defined for all non-negative values of x, not just those greater than 1.
Option B: $x \ \textgreater \ 0$ is not correct, since the quotient is defined for all non-negative values of x, including zero.
Option D: $x \ \textless \ 0$ is not correct, since the quotient is not defined for negative values of x.

Final Thoughts

In conclusion, the inequality that represents all values of x for which the quotient $\sqrt{6 x^2} \div \sqrt{4 x}$ is defined is $x \geq 0$ . This is the most restrictive condition that ensures the quotient is defined, and it's essential to consider both the non-negativity of expressions inside square roots and the condition that the denominator cannot be equal to zero.
Quotient and Inequality: A Q&A Article

Understanding the Problem

In our previous article, we discussed the inequality that represents all values of x for which the quotient $\sqrt{6 x^2} \div \sqrt{4 x}$ is defined. We concluded that the correct answer is $x \geq 0$ . In this article, we'll provide a Q&A section to help clarify any doubts and provide additional insights.

Q: What is the condition for the expression inside the first square root to be non-negative?

A: The expression inside the first square root, $6 x^2$ , is always non-negative for any real value of x, since it's a squared term.

Q: What is the condition for the expression inside the second square root to be non-negative?

A: The expression inside the second square root, $4 x$ , is non-negative if and only if $x \geq 0$ .

Q: What is the condition for the denominator to be non-zero?

A: The denominator of the quotient is $\sqrt{4 x}$ . For the quotient to be defined, the denominator cannot be equal to zero. This means that $4 x \neq 0$ , which implies that $x \neq 0$ .

Q: How do we combine the conditions for non-negativity and the condition that the denominator cannot be equal to zero?

A: Combining the conditions, we get that the quotient is defined if and only if $x \geq 0$ and $x \neq 0$ . Since $x \neq 0$ is implied by $x \geq 0$ , the final condition is simply $x \geq 0$ .

Q: Why is the inequality $x \geq 0$ the most restrictive condition that ensures the quotient is defined?

A: The inequality $x \geq 0$ is the most restrictive condition because the expression inside the first square root is always non-negative, and the denominator cannot be equal to zero. Therefore, the quotient is defined for all non-negative values of x.

Q: How does the inequality $x \geq 0$ compare with the other options?

A: Let's compare the correct answer with the other options:

Option A: $x \ \textgreater \ 1$ is not correct, since the quotient is defined for all non-negative values of x, not just those greater than 1.
Option B: $x \ \textgreater \ 0$ is not correct, since the quotient is defined for all non-negative values of x, including zero.
Option D: $x \ \textless \ 0$ is not correct, since the quotient is not defined for negative values of x.

Q: What is the final answer?

A: The final answer is C. $x \geq 0$ .

Additional Insights

The inequality $x \geq 0$ is the most restrictive condition that ensures the quotient is defined.
The quotient is defined for all non-negative values of x.
The expression inside the first square root is always non-negative, and the denominator cannot be equal to zero.

Conclusion

In this Q&A article, we've provided additional insights and clarification on the inequality that represents all values of x for which the quotient $\sqrt{6 x^2} \div \sqrt{4 x}$ is defined. We've also compared the correct answer with the other options and provided a final answer.