Question

If \(x^{2}

Short answer

Answer: (3) has no value.

Step-by-step solution

Analyze inequality with negative numbers

We are given an inequality, \(x^2 < n\), where \(n\) is a negative number. Let's think about this inequality for a moment. When we square any real number, whether positive, negative, or zero, we get a non-negative value (0 or positive).

Deduce conclusion

Since the square of a real number is always non-negative, it means that if we have a negative number on the right side of the inequality, it is always false. Because, no square of a real number can be less than a negative number.

Choose the correct option

Knowing that no values of x will make the inequality true, the correct answer is: (3) has no value.

Key concepts

Real Numbers

Real numbers are the core of many mathematical concepts. They encompass all numbers on the number line, including positive numbers, negative numbers, and zero. This also covers both rational numbers (like fractions) and irrational numbers (like \( \sqrt{2} \) or \( \pi \)).
Real numbers can be visualized as points on an infinite line, where every point corresponds to a unique real number.

• Rational numbers are real numbers that can be written as a ratio of two integers.
• Irrational numbers are real numbers that cannot be expressed as a simple fraction.

In our context, knowing that \((x^2 < n)\) applies to real numbers is crucial. Since \((n)\) is negative in this exercise, examining real numbers helps us understand that \((x^2)\), when squared, cannot meet these conditions.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign. They are positioned to the left of zero on the number line. Understanding negative numbers is essential because:
Negative numbers have unique properties that affect mathematical operations and inequalities.

• Addition of a negative number is equal to subtraction.
• Multiplication or division of two negative numbers results in a positive number.

In the case of inequalities like \(x^2 < n\), where \(n\) is negative, it is vital to recognize that no squared values of real numbers can be negative, thus shining light on unique qualities of negative numbers.

Squared Values

When a number is squared, it's multiplied by itself, represented as \(x^2\). The results of squaring any real number \(x\) are always non-negative (either zero or positive), and this is due to the property of multiplication:
The product of two positive numbers or two negative numbers is positive. The squaring operation results in absolute value interpretations.

• If x is positive, \(x^2\) is positive.
• If x is negative, \(x^2\) is still positive.
• If x is zero, \(x^2\) equals zero.

In the exercise, the impossibility of having a squared value less than a negative number becomes evident, leading to the conclusion that the inequality offers no solution.

Inequality Analysis

Inequality analysis involves examining mathematical statements where two values are compared using symbols like \(< \, \leq \, > \, \geq\). The exercise features an inequality \(x^2 < n\), offering insights into mathematical analysis:

• Understand that \(n\)'s nature (here, being negative) affects the interpretation.
• Recognize that real numbers squared can never be less than any negative number, as they are always non-negative.
• The comparison provides a context for deducing the absence of solutions based on existing mathematical knowledge.

The analysis concludes by deducing that such inequalities leave no permissible value for \(x\) when \(n\) is negative, thereby arriving at the specific answer choice that x "has no value".