91Ó°ÊÓ

To solve an absolute value equation, the first step is to understand what absolute value means. The absolute value of a number represents its distance from zero on a number line, irrespective of its direction. For example, both 6 and -6 are 6 units away from zero.
Consider the equation \(|p| = 6\). This means we're looking for the numbers whose distance from zero is precisely 6 units. Therefore, there are two possible solutions: \(p = 6\) and \(p = -6\).

For the equation \(|p| = 0\), the absolute value tells us that the distance from zero is 0 units. The only number that fits this requirement is zero itself, so \(p = 0\).Equations where the absolute value is set to a negative number, such as \( |p| = -6 \), are unsolvable. This is because absolute values are always non-negative, meaning they can never be less than zero.

Absolute value has several important properties to keep in mind:

1. An absolute value is always non-negative: \(|p| \geq 0 \) for any real number p. Absolute values measure distance, which can’t be negative.
2. Symmetry: \(|p| = |-p| \). This means the absolute value of a number is the same as the absolute value of its negative counterpart.
3. Zero Property: The absolute value of zero is zero. \(|0| = 0 \), because zero is exactly zero units away from itself.
4. Triangle Inequality: \(|a + b| \leq |a| + |b| \). This property highlights how the absolute value of a sum is at most the sum of the absolute values.

These properties help in understanding and simplifying absolute value expressions.

The most crucial property of absolute values is that they cannot be negative. This is because they measure distance. Distance, by definition, cannot be a negative quantity.

When working with absolute value equations, always remember: any solution must make the absolute value either positive or zero.

For example, solving \(|p| = -6\) is impossible because there is no real number whose distance from zero is a negative value. Thus, always ensure that equations involving absolute values respect this fundamental principle.