Analyzing the Expressions Inside the Square Roots

Oct 13, 2025 by windailys.com 76 views

Understanding the Problem

When dealing with square roots, it's essential to consider the values of the expressions inside the square roots to ensure they are non-negative. This is because the square root of a negative number is undefined in the real number system. Therefore, we need to find the inequality that represents all values of x for which the product $\sqrt{5x} \cdot \sqrt{x+3}$ is defined.

Analyzing the Expressions Inside the Square Roots

To determine the values of x for which the product is defined, we need to consider the expressions inside each square root. The first expression is $\sqrt{5x}$ , and the second expression is $\sqrt{x+3}$ . For both expressions to be defined, the values inside the square roots must be non-negative.

Expression 1: $\sqrt{5x}$

For the expression $\sqrt{5x}$ to be defined, the value inside the square root, which is $5x$ , must be non-negative. This means that $5x \geq 0$ . To solve this inequality, we can divide both sides by 5, which gives us $x \geq 0$ .

Expression 2: $\sqrt{x+3}$

For the expression $\sqrt{x+3}$ to be defined, the value inside the square root, which is $x+3$ , must be non-negative. This means that $x+3 \geq 0$ . To solve this inequality, we can subtract 3 from both sides, which gives us $x \geq -3$ .

Combining the Inequalities

Now that we have analyzed both expressions, we need to combine the inequalities to find the values of x for which the product is defined. The first inequality is $x \geq 0$ , and the second inequality is $x \geq -3$ . Since both inequalities are in the "greater than or equal to" form, we can combine them by finding the intersection of the two sets of values.

The intersection of the two sets of values is the set of values that satisfy both inequalities. In this case, the intersection is the set of values that are greater than or equal to -3 and also greater than or equal to 0. This means that the values of x must be greater than or equal to -3.

Conclusion

Based on our analysis, the inequality that represents all values of x for which the product $\sqrt{5x} \cdot \sqrt{x+3}$ is defined is $x \geq -3$ . This means that the product is defined for all values of x that are greater than or equal to -3.

Final Answer

The final answer is $\boxed{C}$ , which represents the inequality $x \geq -3$ .

Discussion

This problem requires a deep understanding of the properties of square roots and how to analyze expressions inside square roots. It also requires the ability to combine inequalities to find the values of x for which the product is defined.

Example Problems

Find the values of x for which the product $\sqrt{2x} \cdot \sqrt{x-1}$ is defined.
Find the values of x for which the product $\sqrt{x+2} \cdot \sqrt{x-4}$ is defined.

Practice Problems

Find the values of x for which the product $\sqrt{3x} \cdot \sqrt{x+1}$ is defined.
Find the values of x for which the product $\sqrt{x-2} \cdot \sqrt{x+5}$ is defined.

Solutions

The values of x for which the product $\sqrt{2x} \cdot \sqrt{x-1}$ is defined are $x \geq 1$ .
The values of x for which the product $\sqrt{x+2} \cdot \sqrt{x-4}$ is defined are $x \geq 4$ .

Conclusion

In conclusion, the inequality that represents all values of x for which the product $\sqrt{5x} \cdot \sqrt{x+3}$ is defined is $x \geq -3$ . This means that the product is defined for all values of x that are greater than or equal to -3.

Introduction

In our previous article, we discussed the inequality that represents all values of x for which the product $\sqrt{5x} \cdot \sqrt{x+3}$ is defined. We found that the inequality is $x \geq -3$ . In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q: What is the condition for the product of two square roots to be defined?

A: The product of two square roots is defined if and only if the values inside each square root are non-negative.

Q: How do I determine the values of x for which the product of two square roots is defined?

A: To determine the values of x for which the product of two square roots is defined, you need to analyze the expressions inside each square root and find the values of x that make both expressions non-negative.

Q: What is the difference between the inequalities $x \geq 0$ and $x \geq -3$ ?

A: The inequality $x \geq 0$ means that x is greater than or equal to 0, while the inequality $x \geq -3$ means that x is greater than or equal to -3. The first inequality is a subset of the second inequality, meaning that all values of x that satisfy the first inequality also satisfy the second inequality.

Q: Can I combine the inequalities $x \geq 0$ and $x \geq -3$ ?

A: Yes, you can combine the inequalities $x \geq 0$ and $x \geq -3$ by finding the intersection of the two sets of values. The intersection of the two sets of values is the set of values that satisfy both inequalities.

Q: What is the intersection of the inequalities $x \geq 0$ and $x \geq -3$ ?

A: The intersection of the inequalities $x \geq 0$ and $x \geq -3$ is the set of values that are greater than or equal to -3.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{C}$ , which represents the inequality $x \geq -3$ .

Conclusion

Example Problems

Find the values of x for which the product $\sqrt{2x} \cdot \sqrt{x-1}$ is defined.
Find the values of x for which the product $\sqrt{x+2} \cdot \sqrt{x-4}$ is defined.

Practice Problems

Find the values of x for which the product $\sqrt{3x} \cdot \sqrt{x+1}$ is defined.
Find the values of x for which the product $\sqrt{x-2} \cdot \sqrt{x+5}$ is defined.

Solutions

The values of x for which the product $\sqrt{2x} \cdot \sqrt{x-1}$ is defined are $x \geq 1$ .
The values of x for which the product $\sqrt{x+2} \cdot \sqrt{x-4}$ is defined are $x \geq 4$ .