Understanding Absolute Value: The Distance from Zero
The concept of absolute value, denoted as |x - 3|, is fundamental in mathematics, particularly in topics related to algebra and calculus. The absolute value of a number represents its distance from zero on the number line, regardless of its direction (positive or negative).
Definition and Calculation
The absolute value of x - 3 is defined as follows:
|x - 3| begin{cases} x - 3 text{if } x - 3 geq 0 3 - x text{if } x - 3 0 end{cases}
This can be simplified to:
|x - 3| begin{cases} x - 3 text{if } x geq 3 3 - x text{if } x 3 end{cases}
Depending on the value of x, the absolute value will either be x - 3 or 3 - x. This ensures that the result is always non-negative, as the absolute value can never be negative.
Practical Examples
To better understand the application of absolute values, let's consider a few examples.
Example 1: |3 - 4x| 5
Step 1: Isolate the absolute value5 |4x - 3|
Step 2: Remove the absolute value by considering two cases: Case 1:4x - 3 5
4x 8
x 2
Case 2:4x - 3 -5
4x -2
x -0.5
Thus, x can be either 2 or -0.5.
Example 2: |x - 4| 5
Step 1: Isolate the absolute value5 |x - 4|
Step 2: Solve for x considering the two cases: Case 1:x - 4 5
x 9
Case 2:x - 4 -5
x -1
Therefore, x can be either 9 or -1.
Range of Values
The absolute value of a number can range from zero to infinity, just as the number itself can range from minus infinity to plus infinity. This means that for any real number x, the expression |x - 3| will always yield a non-negative result.
Conclusion
The concept of absolute value, or the distance from zero, is a powerful tool in solving various mathematical problems. Understanding the definition and application of absolute values can greatly enhance your problem-solving skills in algebra and beyond.