91Ó°ÊÓ

Absolute value is a fundamental concept in math that represents the distance a number is from zero on the number line, regardless of direction. The absolute value is always non-negative, which means that the absolute value of both positive and negative numbers is positive. For any real number, the notation for absolute value is \(|a|\), where \(|a|\) describes the magnitude of \(a\) without considering its sign. In simple terms:

• If \(a \geq 0\), then \(|a| = a\).
• If \(a < 0\), then \(|a| = -a\).

In the given exercise, you have \(|x| = \frac{1}{x}\) with the condition that \(x > 0\), leading to \(|x| = x\). This simplifies your task as it allows you to directly use \(x\) instead of considering the negative values.

Solving equations that involve absolute values requires a good understanding of algebraic manipulation. In the case of \(x = \frac{1}{x}\), you start by manipulating the equation to get rid of the fraction. This is achieved by multiplying both sides by \(x\), assuming \(x eq 0\):\[x \cdot x = x \cdot \frac{1}{x}\]resulting in \(x^2 = 1\). The next step involves solving this quadratic equation. By taking the square root of both sides, you derive:\[x = \pm1\]Since the context of the problem specifies that \(x > 0\), the negative solution is discarded, leaving \(x = 1\). Therefore, the proper solution given the initial conditions is \(x = 1\). Always remember to check your conditions and constraints when solving similar equations.

Positive numbers are all the numbers greater than zero on the number line. These are the numbers \(\cdots, -3, -2, -1, 0, 1, 2, 3, \cdots\), among which positive numbers are to the right of zero. In problems where \(x > 0\), such as the exercise at hand, we are specifically interested in these positive numbers. They make sense in many real-world scenarios like measurements of height or quantities that cannot be negative.

• Positive numbers are expressed as just numbers, with no sign or sometimes a "+" sign.
• They play a role in equations that demand a solution greater than zero.

In this exercise, we were directed to find a positive \(x\) such that \(|x| = \frac{1}{x}\), and arrived at \(x = 1\). Positive constraints often simplify the options when solving for absolute values, as they immediately limit negative possibilities.