QUESTION 11.1 If X ∈ { − 2 , 2 , 4 , 5 } X \in \{-2, 2, 4, 5\} X ∈ { − 2 , 2 , 4 , 5 } , Choose The Value Of X X X From The Given Set That Will Make The Expression 18 4 − X \sqrt{\frac{18}{4-x}} 4 − X 18 ​ ​ Be:1.1.1 Rational 1.1.2 Undefined 1.1.3 Irrational 1.1.4 Non-real 1.2 Factorize

Introduction

In this problem, we are given an expression $\sqrt{\frac{18}{4-x}}$ and a set of values for $x$. We need to determine which value of $x$ will make the expression rational, undefined, irrational, or non-real. To do this, we will analyze the expression and the given values for $x$.

Understanding the Expression

The expression $\sqrt{\frac{18}{4-x}}$ involves a square root and a fraction. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we are taking the square root of a fraction, which is $\frac{18}{4-x}$. For the expression to be defined, the value inside the square root must be non-negative.

Analyzing the Fraction

The fraction $\frac{18}{4-x}$ is defined as long as the denominator $4-x$ is not equal to zero. This means that $x$ cannot be equal to 4, as it would make the denominator zero and the expression undefined.

Evaluating the Square Root

For the expression to be rational, the value inside the square root must be a perfect square. In this case, the value inside the square root is $\frac{18}{4-x}$. We need to find a value of $x$ that makes this value a perfect square.

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. For example, 4 is a perfect square because it can be expressed as $2^2$. Similarly, 9 is a perfect square because it can be expressed as $3^2$.

Finding the Value of x

To find the value of $x$ that makes the expression rational, we need to find a value of $x$ that makes the value inside the square root a perfect square. Let's analyze the given values for $x$:

• $x = -2$: In this case, the value inside the square root is $\frac{18}{4-(-2)} = \frac{18}{6} = 3$. This is not a perfect square.
• $x = 2$: In this case, the value inside the square root is $\frac{18}{4-2} = \frac{18}{2} = 9$. This is a perfect square because it can be expressed as $3^2$.
• $x = 4$: In this case, the expression is undefined because the denominator is zero.
• $x = 5$: In this case, the value inside the square root is $\frac{18}{4-5} = \frac{18}{-1} = -18$. This is not a perfect square.

Conclusion

Based on our analysis, we can conclude that the value of $x$ that makes the expression rational is $x = 2$. This is because the value inside the square root is a perfect square, specifically $3^2$.

Answer

The value of $x$ that makes the expression rational is $x = 2$.

Factorizing the Expression

To factorize the expression, we need to find the factors of the numerator and the denominator. The numerator is 18, and the denominator is $4-x$.

Factors of the Numerator

The factors of 18 are 1, 2, 3, 6, 9, and 18.

Factors of the Denominator

The factors of $4-x$ depend on the value of $x$. If $x = 2$, then the factors of $4-x$ are 2.

Factorizing the Expression

To factorize the expression, we can write it as:

$$\sqrt{\frac{18}{4-x}} = \sqrt{\frac{2 \cdot 9}{2}} = \sqrt{9} = 3$$

Conclusion

Based on our analysis, we can conclude that the value of $x$ that makes the expression rational is $x = 2$. This is because the value inside the square root is a perfect square, specifically $3^2$. We can also factorize the expression by finding the factors of the numerator and the denominator.

Answer

Q&A

Q: What is the expression $\sqrt{\frac{18}{4-x}}$? A: The expression $\sqrt{\frac{18}{4-x}}$ involves a square root and a fraction. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we are taking the square root of a fraction, which is $\frac{18}{4-x}$.

Q: What is the condition for the expression to be defined? A: For the expression to be defined, the value inside the square root must be non-negative. This means that the denominator $4-x$ must be non-zero.

Q: What is the value of $x$ that makes the expression undefined? A: The value of $x$ that makes the expression undefined is $x = 4$. This is because the denominator $4-x$ becomes zero when $x = 4$, making the expression undefined.

Q: What is the value of $x$ that makes the expression rational? A: The value of $x$ that makes the expression rational is $x = 2$. This is because the value inside the square root is a perfect square, specifically $3^2$.

Q: How can we factorize the expression? A: To factorize the expression, we need to find the factors of the numerator and the denominator. The numerator is 18, and the denominator is $4-x$. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of $4-x$ depend on the value of $x$. If $x = 2$, then the factors of $4-x$ are 2.

Q: What is the final value of the expression when $x = 2$? A: The final value of the expression when $x = 2$ is $\sqrt{9} = 3$.

Q: What is the nature of the expression when $x = 2$? A: The expression is rational when $x = 2$.

Q: What is the nature of the expression when $x = 4$? A: The expression is undefined when $x = 4$.

Q: What is the nature of the expression when $x = 5$? A: The expression is irrational when $x = 5$.

Conclusion

In this article, we have evaluated the expression $\sqrt{\frac{18}{4-x}}$ and determined the value of $x$ that makes the expression rational, undefined, irrational, or non-real. We have also factorized the expression and found the final value of the expression when $x = 2$.

Answer

The value of $x$ that makes the expression rational is $x = 2$. The expression is undefined when $x = 4$. The expression is irrational when $x = 5$.

Final Thoughts

Evaluating expressions with square roots and fractions can be challenging, but by understanding the conditions for the expression to be defined and the nature of the expression, we can determine the value of $x$ that makes the expression rational, undefined, irrational, or non-real.