QUESTION 11.1 If $x \in {-2, 2, 4, 5}$, Choose The Value Of $x$ From The Given Set That Will Make The Expression $\sqrt{\frac{18}{4-x}}$ Be:- 1.1.1 Rational- 1.1.2 Undefined- 1.1.3 Irrational- 1.1.4 Non-real1.2

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Evaluating the Expression: Rational, Undefined, Irrational, or Non-Real?

Understanding the Problem

When dealing with mathematical expressions, it's essential to consider the properties of the numbers involved. In this case, we're given an expression $\sqrt{\frac{18}{4-x}}$ and a set of possible values for $x$: ${-2, 2, 4, 5}$. Our task is to determine which value of $x$ will make the expression rational, undefined, irrational, or non-real.

Analyzing the Expression

To begin, let's break down the expression $\sqrt{\frac{18}{4-x}}$. We have a square root of a fraction, where the numerator is a constant (18) and the denominator is a variable expression ($4-x$). For the expression to be defined, the denominator must be non-zero, i.e., $4-x \neq 0$.

Evaluating the Denominator

Now, let's examine the values of $x$ from the given set and determine which one makes the denominator zero.

  • For $x = -2$, we have $4 - (-2) = 6 \neq 0$.
  • For $x = 2$, we have $4 - 2 = 2 \neq 0$.
  • For $x = 4$, we have $4 - 4 = 0$.
  • For $x = 5$, we have $4 - 5 = -1 \neq 0$.

As we can see, only $x = 4$ makes the denominator zero, which means the expression is undefined for this value of $x$.

Evaluating the Expression for the Remaining Values

Now that we've identified the value of $x$ that makes the expression undefined, let's examine the remaining values.

  • For $x = -2$, we have $\sqrt{\frac{18}{6}} = \sqrt{3}$, which is an irrational number.
  • For $x = 2$, we have $\sqrt{\frac{18}{2}} = \sqrt{9} = 3$, which is a rational number.
  • For $x = 5$, we have $\sqrt{\frac{18}{-1}} = \sqrt{-18}$, which is a non-real number (or imaginary number).

Conclusion

In conclusion, the value of $x$ that will make the expression $\sqrt{\frac{18}{4-x}}$ rational is $x = 2$, as it results in a rational number (3). The value of $x$ that will make the expression undefined is $x = 4$, as it results in a zero denominator. The values of $x$ that will make the expression irrational are $x = -2$ and $x = 5$, as they result in irrational numbers. Finally, the value of $x$ that will make the expression non-real is $x = 5$, as it results in a non-real number.

Key Takeaways

  • When dealing with mathematical expressions, it's essential to consider the properties of the numbers involved.
  • The value of $x$ that makes the expression undefined is $x = 4$.
  • The values of $x$ that make the expression rational, irrational, or non-real are $x = 2$, $x = -2$, and $x = 5$, respectively.

Further Exploration

This problem is an excellent example of how mathematical expressions can be evaluated based on the properties of the numbers involved. It's essential to understand the concepts of rational, irrational, and non-real numbers, as well as the properties of square roots and fractions. By exploring these concepts, you'll develop a deeper understanding of mathematical expressions and be better equipped to tackle complex problems.
Evaluating Mathematical Expressions: A Q&A Guide

Introduction

In our previous article, we explored the concept of evaluating mathematical expressions, specifically the expression $\sqrt{\frac{18}{4-x}}$. We determined that the value of $x$ that makes the expression undefined is $x = 4$, and the values of $x$ that make the expression rational, irrational, or non-real are $x = 2$, $x = -2$, and $x = 5$, respectively. In this article, we'll provide a Q&A guide to help you better understand the concepts and techniques involved in evaluating mathematical expressions.

Q: What is the difference between a rational and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. An irrational number, on the other hand, is a number that cannot be expressed as a ratio of two integers. Examples of irrational numbers include $\sqrt{2}$ and $\pi$.

Q: How do I determine if a number is rational or irrational?

A: To determine if a number is rational or irrational, you can try to express it as a ratio of two integers. If you can express the number as a ratio of two integers, it is rational. If you cannot express the number as a ratio of two integers, it is irrational.

Q: What is the difference between a real and a non-real number?

A: A real number is a number that can be expressed on the number line, i.e., it is a number that can be measured or counted. A non-real number, on the other hand, is a number that cannot be expressed on the number line, i.e., it is a number that is not measurable or countable. Examples of non-real numbers include imaginary numbers, such as $i$ and $-i$.

Q: How do I determine if a number is real or non-real?

A: To determine if a number is real or non-real, you can try to express it on the number line. If you can express the number on the number line, it is real. If you cannot express the number on the number line, it is non-real.

Q: What is the difference between a square root and a fraction?

A: A square root is a mathematical operation that finds the number that, when multiplied by itself, gives a specified value. A fraction, on the other hand, is a mathematical expression that represents a part of a whole. Examples of fractions include $\frac{1}{2}$ and $\frac{3}{4}$.

Q: How do I evaluate a square root expression?

A: To evaluate a square root expression, you can follow these steps:

  1. Simplify the expression inside the square root, if possible.
  2. Find the square root of the simplified expression.
  3. Simplify the resulting expression, if possible.

Q: How do I evaluate a fraction expression?

A: To evaluate a fraction expression, you can follow these steps:

  1. Simplify the numerator and denominator, if possible.
  2. Divide the numerator by the denominator.
  3. Simplify the resulting expression, if possible.

Q: What are some common mistakes to avoid when evaluating mathematical expressions?

A: Some common mistakes to avoid when evaluating mathematical expressions include:

  • Not simplifying the expression inside the square root or fraction.
  • Not following the order of operations (PEMDAS).
  • Not checking for undefined values.
  • Not simplifying the resulting expression.

Conclusion

Evaluating mathematical expressions is an essential skill in mathematics, and it requires a deep understanding of the concepts and techniques involved. By following the steps outlined in this article, you'll be better equipped to evaluate mathematical expressions and avoid common mistakes. Remember to always simplify the expression inside the square root or fraction, follow the order of operations (PEMDAS), check for undefined values, and simplify the resulting expression. With practice and patience, you'll become proficient in evaluating mathematical expressions and tackle complex problems with confidence.