Denominator Cannot Be Equal to Zero

Oct 13, 2025 by recoveryshake.com 145 views

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

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. This involves analyzing the properties of square roots and the conditions under which the expression is valid.

The given quotient is $\frac{\sqrt{7 x^2}}{\sqrt{3 x}}$ . For this expression to be defined, the denominator cannot be equal to zero, and the square roots must be real numbers. Let's break down the conditions for each of these requirements.

Denominator Cannot Be Equal to Zero

The denominator of the quotient is $\sqrt{3 x}$ . For this expression to be defined, the value inside the square root must be non-negative. This means that $3x \geq 0$ . Since $3$ is a positive constant, the only way for this inequality to hold is if $x \geq 0$ .

Square Roots Must Be Real Numbers

The square roots in the expression are $\sqrt{7 x^2}$ and $\sqrt{3 x}$ . For these expressions to be real numbers, the values inside the square roots must be non-negative. This means that $7x^2 \geq 0$ and $3x \geq 0$ . Since $7$ and $3$ are positive constants, the only way for these inequalities to hold is if $x^2 \geq 0$ and $x \geq 0$ . However, since $x^2 \geq 0$ is always true for any real number $x$ , we can focus on the condition $x \geq 0$ .

Based on the analysis above, the inequality that represents all values of $x$ for which the quotient is defined is $x \geq 0$ . This is because the denominator cannot be equal to zero, and the square roots must be real numbers, which requires $x \geq 0$ .

Let's compare the options given in the problem:

A. $x \ \textgreater \ -1$ : This option is incorrect because it does not guarantee that the denominator is non-zero.
B. $x \ \textgreater \ 0$ : This option is correct because it guarantees that the denominator is non-zero and the square roots are real numbers.
C. $x \geq 0$ : This option is also correct because it guarantees that the denominator is non-zero and the square roots are real numbers.

In the previous article, we discussed the inequality that represents all values of $x$ for which the quotient $\frac{\sqrt{7 x^2}}{\sqrt{3 x}}$ is defined. 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 denominator to be non-zero?

A: The denominator is $\sqrt{3 x}$ . For this expression to be defined, the value inside the square root must be non-negative. This means that $3x \geq 0$ . Since $3$ is a positive constant, the only way for this inequality to hold is if $x \geq 0$ .

Q: What is the condition for the square roots to be real numbers?

A: The square roots in the expression are $\sqrt{7 x^2}$ and $\sqrt{3 x}$ . For these expressions to be real numbers, the values inside the square roots must be non-negative. This means that $7x^2 \geq 0$ and $3x \geq 0$ . Since $7$ and $3$ are positive constants, the only way for these inequalities to hold is if $x^2 \geq 0$ and $x \geq 0$ . However, since $x^2 \geq 0$ is always true for any real number $x$ , we can focus on the condition $x \geq 0$ .

Q: Why is option A incorrect?

A: Option A is incorrect because it does not guarantee that the denominator is non-zero. The inequality $x \ \textgreater \ -1$ does not ensure that $3x \geq 0$ , which is a necessary condition for the denominator to be non-zero.

Q: Why is option C correct?

A: Option C is correct because it guarantees that the denominator is non-zero and the square roots are real numbers. The inequality $x \geq 0$ ensures that $3x \geq 0$ , which means the denominator is non-zero. Additionally, $x \geq 0$ ensures that the square roots are real numbers.

Q: What is the final answer?

A: The final answer is $\boxed{B}$ , which represents the inequality $x \ \textgreater \ 0$ . This is the correct inequality that represents all values of $x$ for which the quotient is defined.

The quotient is defined when the denominator is non-zero and the square roots are real numbers.
The condition for the denominator to be non-zero is $x \geq 0$ .
The condition for the square roots to be real numbers is also $x \geq 0$ .
Option A is incorrect because it does not guarantee that the denominator is non-zero.
Option C is correct because it guarantees that the denominator is non-zero and the square roots are real numbers.

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 is defined. We've also addressed common questions and provided a final answer.