Example: Determine a Number within a Prescribed Distance

Describe all values [latex]x[/latex] within or including a distance of 4 from the number 5.

Answer: We want the distance between [latex]x[/latex] and 5 to be less than or equal to 4. We can draw a number line to represent the condition to be satisfied. The distance from [latex]x[/latex] to 5 can be represented using [latex]|x - 5|[/latex]. We want the values of [latex]x[/latex] that satisfy the condition [latex]|x - 5|\le 4[/latex].

Analysis of the Solution

Note that [latex-display]\displaystyle{-4}\le{x - 5}[/latex-display] [latex-display]\displaystyle{1}\le{x}[/latex-display] And: [latex-display]\displaystyle{x-5}\le{4}[/latex-display] [latex-display]\displaystyle{x}\le{9}[/latex-display] So [latex]|x - 5|\le 4[/latex] is equal to [latex]1\le x\le 9[/latex]. However, mathematicians generally prefer absolute value notation.

Example: 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 [latex]\pm 1\%,\pm5\%,[/latex] or [latex]\displaystyle\pm10\%[/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.

Answer: 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 [latex]R[/latex] in ohms,

[latex]|R - 680|\le 34[/latex]

Example: Writing an Equation for an Absolute Value Function

Write an equation for the function graphed below.

Answer: The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance. From this information we can write the equation

[latex]\begin{array}{l}f\left(x\right)=2\left|x - 3\right|-2,\hfill & \text{treating the stretch as a vertical stretch, or}\hfill \\ f\left(x\right)=\left|2\left(x - 3\right)\right|-2,\hfill & \text{treating the stretch as a horizontal compression}.\hfill \end{array}[/latex]

Analysis of the Solution

Note that these equations are algebraically the same—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.

Example: Finding the Zeros of an Absolute Value Function

For the function [latex]f\left(x\right)=|4x+1|-7[/latex] , find the values of [latex]x[/latex] such that [latex]\text{ }f\left(x\right)=0[/latex] .

Answer:

[latex]\begin{array}{l}0=|4x+1|-7\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Substitute 0 for }f\left(x\right).\hfill \\ 7=|4x+1|\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Isolate the absolute value on one side of the equation}.\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ 7=4x+1\hfill & \text{or}\hfill & \hfill & \hfill & \hfill & -7=4x+1\hfill & \text{Break into two separate equations and solve}.\hfill \\ 6=4x\hfill & \hfill & \hfill & \hfill & \hfill & -8=4x\hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ x=\frac{6}{4}=1.5\hfill & \hfill & \hfill & \hfill & \hfill & \text{ }x=\frac{-8}{4}=-2\hfill & \hfill \end{array}[/latex]

The function outputs 0 when [latex]x=1.5[/latex] or [latex]x=-2[/latex].