Understanding the Absolute Value Function: A Mathematical Exploration of Non-Negative Values and Distance on the Number Line

Absolute Value Function

The absolute value function, often denoted as |x|, is a mathematical function that returns the non-negative value of a given number

The absolute value function, often denoted as |x|, is a mathematical function that returns the non-negative value of a given number. In other words, it gives the distance of the number from zero on the number line.

The absolute value function can be defined as follows:
For any real number x, the absolute value of x, |x|, is equal to x if x is greater than or equal to zero. If x is less than zero, then |x| is equal to the negative of x.

To better understand how the absolute value function works, let’s consider a few examples:

Example 1:
Find the absolute value of -5.
Since -5 is less than zero, we take the negative of -5. Therefore, | -5 | = -(-5) = 5.
So, the absolute value of -5 is 5.

Example 2:
Find the absolute value of 2.
Since 2 is greater than or equal to zero, we don’t need to take the negative. Thus, | 2 | = 2.
So, the absolute value of 2 is 2.

Example 3:
Find the value of x if |x| = 7.
In this case, we have an equation involving the absolute value function. To solve it, we consider two cases:

Case 1: x is positive or zero
If x is greater than or equal to zero, then |x| = x. So, we have x = 7.

Case 2: x is negative
If x is less than zero, then |x| = -x. So, we have -x = 7.
To isolate x, we multiply both sides of the equation by -1 to get x = -7.

In this example, we find that the value of x could be either 7 or -7, depending on whether it is positive or negative.

The absolute value function is frequently used in various mathematical concepts, including solving equations, inequalities, and distance-related problems. It ensures that distances are always positive while disregarding the direction.

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