The absolute value of a number represents its distance from zero on the number line. It is always non-negative, regardless of whether the original number is positive or negative. For example, both 3 and negative 3 have an absolute value of 3, because they are both 3 units away from zero.
The absolute value function has several key properties. First, it is always non-negative. Second, the absolute value of a number equals the absolute value of its negative. The graph of y equals absolute value of x forms a V-shape, symmetric about the y-axis, with its vertex at the origin.
When solving absolute value equations like the absolute value of x minus 3 equals 5, we need to consider two cases. The expression inside the absolute value can be either positive or negative. This gives us x minus 3 equals 5, so x equals 8, or x minus 3 equals negative 5, so x equals negative 2. Both solutions are 5 units away from 3 on the number line.
Absolute value inequalities come in two main types. When the absolute value of x is less than a positive number a, the solution is the interval from negative a to positive a. When the absolute value of x is greater than a, the solution consists of two rays: x less than negative a or x greater than positive a. These represent distances from zero on the number line.