For What Values Of The Variable Z Is The Expression $\sqrt{\frac{16-2z}{27}}$ Meaningful? Choose The Correct Answer Option.

In the realm of algebra, understanding the domain of an expression is paramount to ensuring mathematical validity. The domain refers to the set of all possible input values (often represented by variables) for which the expression produces a real and defined output. When dealing with expressions involving square roots, the domain is particularly important because the square root of a negative number is not defined within the set of real numbers. This article delves into the intricacies of determining the values of a variable that make an expression involving a square root meaningful. We will dissect the expression $\sqrt{\frac{16-2z}{27}}$ to ascertain the valid range of values for the variable z. This exploration will empower you with the knowledge to tackle similar problems and foster a deeper comprehension of algebraic expressions. In this comprehensive guide, we will explore the conditions under which an expression involving a square root is meaningful, focusing on the specific case of $\sqrt{\frac{16-2z}{27}}$. Understanding the concept of domain is crucial in algebra, and this article aims to provide a clear and detailed explanation of how to determine the valid values for the variable z in this expression. We'll break down the problem step by step, ensuring you grasp the underlying principles and can apply them to similar scenarios. Whether you're a student learning algebra for the first time or someone looking to refresh your knowledge, this guide will equip you with the necessary tools to confidently tackle domain-related questions. The expression $\sqrt{\frac{16-2z}{27}}$ presents an interesting challenge in determining the meaningful values of the variable z. The presence of a square root introduces a critical constraint: the radicand (the expression under the square root) must be non-negative. This condition stems from the fact that the square root of a negative number is not defined in the set of real numbers. Therefore, our task is to find the values of z that ensure the radicand, $\frac{16-2z}{27}$, is greater than or equal to zero. This involves solving an inequality, which will reveal the range of permissible values for z. By meticulously analyzing this inequality, we can pinpoint the exact interval where the expression is defined. This process not only helps us solve this specific problem but also enhances our understanding of how to handle expressions involving radicals and fractions. The following sections will guide you through the step-by-step solution, emphasizing the key concepts and techniques involved.

Understanding the Square Root Constraint

The core concept in determining the domain of expressions involving square roots lies in the non-negativity requirement of the radicand. This means that the expression under the square root must be greater than or equal to zero. Mathematically, for an expression of the form $\sqrt{f(x)}$, the domain is restricted to the values of x for which $f(x) \ge 0$. This constraint arises from the fundamental definition of the square root function within the realm of real numbers. The square root of a negative number is not a real number; it ventures into the territory of complex numbers, which are beyond the scope of this discussion. To illustrate this point, consider the square root of 9, which is 3 because 3 * 3 = 9. Similarly, the square root of 0 is 0 because 0 * 0 = 0. However, there is no real number that, when multiplied by itself, yields a negative number. This is why we must ensure that the radicand is non-negative. In the given expression, $\sqrt{\frac{16-2z}{27}}$, the radicand is the fraction $\frac{16-2z}{27}$. Therefore, we need to find the values of z that satisfy the inequality $\frac{16-2z}{27} \ge 0$. This inequality is the key to unlocking the domain of the expression. Solving it will reveal the set of all z values for which the expression is defined. This constraint is not merely a mathematical technicality; it has deep implications in various scientific and engineering applications where square roots are used to model real-world phenomena. For instance, in physics, the speed of an object might be represented by a square root expression. In such cases, ensuring the radicand is non-negative guarantees that the calculated speed is a real and physically meaningful quantity. Similarly, in engineering, calculations involving stress and strain often involve square roots, and the non-negativity constraint ensures that the results are within the bounds of physical reality. Understanding this constraint is therefore not just about solving algebraic problems; it's about ensuring the validity and applicability of mathematical models in diverse fields. By mastering this concept, you gain a more profound appreciation for the role of mathematics in describing and understanding the world around us.

Solving the Inequality

To determine the values of z for which the expression $\sqrt{\frac{16-2z}{27}}$ is meaningful, we need to solve the inequality $\frac{16-2z}{27} \ge 0$. The first step in solving this inequality is to eliminate the denominator. Since 27 is a positive number, multiplying both sides of the inequality by 27 will not change the direction of the inequality. This gives us: $16 - 2z \ge 0$. Next, we need to isolate the term containing z. We can do this by subtracting 16 from both sides of the inequality: $-2z \ge -16$. Now, we need to solve for z. To do this, we divide both sides of the inequality by -2. However, it is crucial to remember that dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign. Therefore, when we divide by -2, the \ge sign becomes \le: $z \le 8$. This is the solution to the inequality. It tells us that the expression $\sqrt{\frac{16-2z}{27}}$ is defined for all values of z that are less than or equal to 8. In other words, the domain of the expression is the interval $(-\infty, 8]$. This solution highlights the importance of paying close attention to the rules of inequality manipulation, especially when dealing with negative numbers. A common mistake is to forget to reverse the inequality sign when dividing or multiplying by a negative number, which would lead to an incorrect solution. The solution $z \le 8$ can be visualized on a number line. It represents all the points on the number line that are to the left of 8, including 8 itself. This graphical representation provides a clear understanding of the range of values for which the expression is defined. To further solidify your understanding, it's helpful to test values of z that satisfy the inequality and values that do not. For example, if we let z = 0, which is less than 8, the radicand becomes $\frac{16-2(0)}{27} = \frac{16}{27}$, which is positive, and the square root is defined. On the other hand, if we let z = 9, which is greater than 8, the radicand becomes $\frac{16-2(9)}{27} = \frac{-2}{27}$, which is negative, and the square root is not defined in the real number system. This testing approach helps to confirm the validity of the solution and reinforces the concept of domain. The process of solving this inequality not only provides the answer to this specific problem but also demonstrates a general technique for finding the domain of expressions involving square roots and fractions. This technique can be applied to a wide range of similar problems, making it a valuable tool in your algebraic toolkit.

Conclusion

In conclusion, the expression $\sqrt{\frac{16-2z}{27}}$ is meaningful for all values of z that satisfy the inequality $z \le 8$. This determination was made by understanding the fundamental constraint that the radicand of a square root must be non-negative. By setting up and solving the inequality $\frac{16-2z}{27} \ge 0$, we were able to identify the precise range of values for z that ensure the expression yields a real number. This process involved several key steps, including eliminating the denominator, isolating the term containing z, and remembering to reverse the inequality sign when dividing by a negative number. The solution $z \le 8$ represents the domain of the expression, indicating that any value of z less than or equal to 8 will result in a defined square root. Conversely, any value of z greater than 8 will lead to a negative radicand, rendering the expression undefined within the realm of real numbers. This exercise underscores the importance of understanding the concept of domain in algebra. The domain of an expression is not merely a theoretical concept; it has practical implications in various mathematical and scientific contexts. It ensures that the expressions we work with produce meaningful and valid results. Furthermore, the techniques used to determine the domain in this case are applicable to a broader range of algebraic problems. The ability to solve inequalities and manipulate expressions involving square roots is a valuable skill that will serve you well in your mathematical journey. This article has provided a detailed and step-by-step guide to finding the domain of an expression involving a square root. By understanding the underlying principles and applying the techniques discussed, you can confidently tackle similar problems and deepen your understanding of algebra. Remember, the key to success in mathematics lies in a solid grasp of the fundamental concepts and the ability to apply them effectively. This exploration of the domain of $\sqrt{\frac{16-2z}{27}}$ is a testament to the power of these principles and their role in solving algebraic challenges.