Question

If \(\mathrm{x}^{2}>0\), then find the range of the values that \(\mathrm{x}\) can take. (1) \(\mathrm{x}=0\) (2) \(\mathrm{x} \in \mathrm{R}\) (3) \(\mathrm{x} \in(0, \infty)\) (4) \(\mathrm{x} \in \mathrm{R}-\\{0\\}\)

Short answer

Answer: The range of values for x that satisfy the inequality is \(x \in R - \{0\}\) (x is any real number excluding zero).

Step-by-step solution

Identify the inequality

We are given the inequality \(x^2 > 0\).

Determine the domain of x

We need to find the values of x for which the inequality holds true.

Analyze option (1)

Option (1) states that x = 0. If we plug this value into the inequality: $$(0)^2 > 0$$ $$0 > 0$$ This is not true. So, option (1) is incorrect.

Analyze option (2)

Option (2) states that \(x \in R\) (x is any real number). If x is a positive number, then \(x^2 > 0\) holds true. If x is a negative number, then \(x^2 > 0\) also holds true (since the square of a negative number is positive). However, if x = 0, as we have already seen, the inequality is not true. Therefore, option (2) is also incorrect.

Analyze option (3)

Option (3) states that \(x \in (0, \infty)\) (x is a positive number). If x is a positive number, then \(x^2 > 0\) holds true. Therefore, (3) is a possible solution.

Analyze option (4)

Option (4) states that \(x \in R - \{0\}\) (x is any real number excluding zero). This includes all positive and negative numbers, but not zero. We already know that for positive and negative numbers, \(x^2 > 0\) holds true. Therefore, option (4) is also a possible solution.

Compare options (3) and (4)

Both options (3) and (4) hold true for inequalities involving \(x^2 > 0\). However, option (4) is more general as it includes both positive and negative values, whereas option (3) only includes positive values. Therefore, the correct range for the values of x is given by option (4). So, the range of values that x can take is \(x \in R - \{0\}\).

Key concepts

Real Numbers

Real numbers encompass a vast set of numbers you encounter in everyday life. They include:

• Whole numbers like 0, 1, 2, 3...
• Fractions like 1/2, 3/4...
• Irrational numbers like \(\pi\) and \(\sqrt{2}\)
• Negative numbers like -1, -2...

Real numbers are used in nearly every aspect of our lives. They form a continuous line on a number line with no gaps. When we consider inequalities involving real numbers, such as \(x^2 > 0\), we explore potential values that the real number \(x\) can assume.
Understanding that real numbers include both positive and negative values except zero helps you determine solutions for inequalities accurately.

Square of a Number

The square of a number is the product of a number with itself. For any real number \(x\), the square is \(x^2\). Key properties include:

• The square of any real number is always non-negative \((x^2 \geq 0)\).
• Negative numbers squared become positive – for example, \((-3)^2 = 9\).
• The square of zero is zero \((0^2 = 0)\).

Given these properties, when we deal with \(x^2 > 0\), we can derive that \(x\) must exclude zero, as it is the only number whose square is not strictly positive.
This exercise serves to reinforce your understanding of these properties and their relevance in solving inequalities.

Range of Values

In the context of inequalities, the range of values refers to all the possible real numbers that satisfy the given condition. For \(x^2 > 0\), the ranges can be explored:

• Option (3): \(x \in (0, \infty)\) includes only positive numbers.
• Option (4): \(x \in \mathbb{R} - \{0\}\), which includes all positive and negative numbers, excluding zero.

Comparing these, option (4) is more inclusive since it covers all real numbers except zero. This reveals that the range for \(x\) satisfying the inequality \(x^2 > 0\) cannot include zero, but may include any positive or negative value.

Positive and Negative Numbers

Understanding positive and negative numbers is crucial in solving inequalities like \(x^2 > 0\). Positive numbers are greater than zero \((x > 0)\), and negative numbers are less than zero \((x < 0)\).

• Positive numbers remain positive when squared. For example, \(2^2 = 4\).
• Negative numbers also turn positive when squared, as seen with \((-2)^2 = 4\).

This concept highlights why both positive and negative numbers satisfy \(x^2 > 0\), neither produces non-positive results. Thus, the inequality remains true as long as \(x\) is not zero. Recognizing how these numbers behave when squared helps solve and understand inequalities more effectively.