Recall that in its basic form
A General Note: Absolute Value Function:
The absolute value function can be defined as a piecewise function
Example 1
Problem 1
Determine a Number within a Prescribed Distance
Describe all values
Solution
We want the distance between
The distance from
Try It:
Exercise 1
Describe all values
Solution
Example 2
Problem 1
Resistance of a Resistor
Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often
Suppose we have a resistor rated at 680 ohms,
Solution
5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance
Try It:
Exercise 2
Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.
Solution
using the variable
Analysis
Note that
So

x − 5  ≤ 4

x − 5  ≤ 4 is equivalent to
1 ≤ x ≤ 9.
1 ≤ x ≤ 9.
However, mathematicians generally prefer absolute value notation.