- Graph an absolute value function.
- Solve an absolute value equation.
- Solve an absolute value inequality.
Understanding Absolute Value
Recall that in its basic form [latex]\displaystyle{f}\left({x}\right)={|x|}[/latex], the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.
absolute value
The absolute value of a number is a measure of its distance from zero on the number line, regardless of direction. It is always a non-negative value.
For a real number [latex]x[/latex], the absolute value is denoted by [latex]|x|[/latex] and is defined as:
[latex]|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}[/latex]
Describe all values [latex]x[/latex] within or including a distance of 4 from the number 5.
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 [latex]\displaystyle\pm\text{1%,}\pm\text{5%,}[/latex] or [latex]\displaystyle\pm\text{10%}[/latex].
Suppose we have a resistor rated at 680 ohms, [latex]\pm 5%[/latex]. Use the absolute value function to express the range of possible values of the actual resistance.
The absolute value function can be defined as a piecewise function
[latex]f(x) = \begin{cases} x ,\ x \geq 0 \\ -x , x < 0 \end{cases}[/latex]