Which Inequality Represents All Values Of { X $}$ For Which The Product Below Is Defined? { \sqrt{5x} \cdot \sqrt{x+4}$}$A. { X \ \textgreater \ 0$}$B. { X \geq 0$}$C. { X \leq -4$} D . \[ D. \[ D . \[ X \geq

Mar 9, 2025 by ADMIN 211 views

Understanding the Conditions for the Product of Square Roots to be Defined

When dealing with the product of square roots, it's essential to consider the conditions under which the expression is defined. In this case, we have the product of two square roots: $\sqrt{5x} \cdot \sqrt{x+4}$ . To determine the values of $x$ for which this product is defined, we need to examine the properties of square roots and the conditions that must be met for the expression to be valid.

The Properties of Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, this is represented as $\sqrt{x} = y \Rightarrow y^2 = x$ . For a square root to be defined, the radicand (the number inside the square root) must be non-negative. This means that the value inside the square root must be greater than or equal to zero.

The Conditions for the Product of Square Roots to be Defined

In the case of the product $\sqrt{5x} \cdot \sqrt{x+4}$ , we need to consider the conditions under which both square roots are defined. For the first square root, $\sqrt{5x}$ , the radicand $5x$ must be non-negative. This implies that $5x \geq 0$ , which can be rewritten as $x \geq 0$ . For the second square root, $\sqrt{x+4}$ , the radicand $x+4$ must also be non-negative. This implies that $x+4 \geq 0$ , which can be rewritten as $x \geq -4$ .

Combining the Conditions

To determine the values of $x$ for which the product $\sqrt{5x} \cdot \sqrt{x+4}$ is defined, we need to combine the conditions for both square roots. Since both conditions must be met simultaneously, we take the intersection of the two conditions. This gives us the combined condition: $x \geq 0$ and $x \geq -4$ . Since both conditions are already satisfied by the first condition ( $x \geq 0$ ), we can simplify the combined condition to $x \geq 0$ .

Conclusion

Answer

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

Additional Considerations

It's worth noting that the inequality $x \geq 0$ is the most restrictive condition that satisfies both square roots. This is because the condition $x \geq -4$ is already satisfied by the condition $x \geq 0$ . Therefore, the inequality $x \geq 0$ is the only condition that needs to be considered.

Real-World Applications

The concept of square roots and the conditions for their definition have numerous real-world applications. For example, in physics, the square root of a quantity represents the magnitude of a vector. In engineering, the square root of a quantity represents the magnitude of a force or a stress. In finance, the square root of a quantity represents the volatility of a stock or a portfolio.

Conclusion

In conclusion, the inequality that represents all values of $x$ for which the product $\sqrt{5x} \cdot \sqrt{x+4}$ is defined is $x \geq 0$ . This is because both square roots must be defined, and the conditions for both square roots are met when $x \geq 0$ . The concept of square roots and the conditions for their definition have numerous real-world applications, and understanding these concepts is essential for solving problems in mathematics, physics, engineering, and finance.

References

[1] "Square Roots" by Math Open Reference. Retrieved from https://www.mathopenref.com/sqrt.html
[2] "Properties of Square Roots" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f6b7d/square-roots/x2f6b7e/properties-of-square-roots
[3] "Real-World Applications of Square Roots" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/Real-WorldApplicationsofSquareRoots.html
Frequently Asked Questions (FAQs) about Square Roots and Inequalities

Q: What is the definition of a square root?

A: A square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, this is represented as $\sqrt{x} = y \Rightarrow y^2 = x$ .

Q: What are the conditions for a square root to be defined?

A: For a square root to be defined, the radicand (the number inside the square root) must be non-negative. This means that the value inside the square root must be greater than or equal to zero.

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 examine the properties of square roots and the conditions that must be met for the expression to be valid. You need to consider the conditions for both square roots and take the intersection of the two conditions.

Q: What is the inequality that represents all values of x for which the product $\sqrt{5x} \cdot \sqrt{x+4}$ is defined?

A: The inequality that represents all values of x for which the product $\sqrt{5x} \cdot \sqrt{x+4}$ is defined is $x \geq 0$ .

Q: Why is the inequality $x \geq 0$ the most restrictive condition that satisfies both square roots?

A: The inequality $x \geq 0$ is the most restrictive condition that satisfies both square roots because the condition $x \geq -4$ is already satisfied by the condition $x \geq 0$ . Therefore, the inequality $x \geq 0$ is the only condition that needs to be considered.

Q: What are some real-world applications of square roots and inequalities?

A: The concept of square roots and inequalities has numerous real-world applications. For example, in physics, the square root of a quantity represents the magnitude of a vector. In engineering, the square root of a quantity represents the magnitude of a force or a stress. In finance, the square root of a quantity represents the volatility of a stock or a portfolio.

Q: How can I use square roots and inequalities to solve problems in mathematics, physics, engineering, and finance?

A: To use square roots and inequalities to solve problems in mathematics, physics, engineering, and finance, you need to understand the properties of square roots and the conditions that must be met for the expression to be valid. You need to be able to determine the values of x for which the product of two square roots is defined and use this information to solve problems.

Q: What are some common mistakes to avoid when working with square roots and inequalities?

A: Some common mistakes to avoid when working with square roots and inequalities include:

Not considering the conditions for both square roots
Not taking the intersection of the two conditions
Not understanding the properties of square roots
Not being able to determine the values of x for which the product of two square roots is defined

Q: How can I practice using square roots and inequalities to solve problems?

A: To practice using square roots and inequalities to solve problems, you can try the following:

Work on problems that involve the product of two square roots
Practice determining the values of x for which the product of two square roots is defined
Use online resources, such as Khan Academy and Wolfram MathWorld, to learn more about square roots and inequalities
Practice solving problems in mathematics, physics, engineering, and finance that involve square roots and inequalities

Conclusion

In conclusion, understanding the properties of square roots and the conditions that must be met for the expression to be valid is essential for solving problems in mathematics, physics, engineering, and finance. By practicing using square roots and inequalities to solve problems, you can develop the skills and knowledge you need to succeed in these fields.