Figure 2

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 the absolute value as [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

So [latex]|x - 5|\le 4[/latex] is equivalent to [latex]1\le x\le 9[/latex].

However, mathematicians generally prefer absolute value notation.

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,

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.

Figure 6

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.

Figure 7

From this information we can write the equation

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

Analysis of the Solution

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

[latex]\begin{align}&0=|4x+1|-7 &&&&&& \text{Substitute 0 for }f\left(x\right). \\ &7=|4x+1| &&&&&& \text{Isolate the absolute value on one side of the equation}.\\ & \\ &7=4x+1 & \text{or} &&& -7=4x+1 && \text{Break into two separate equations and solve}. \\ &6=4x &&&& -8=4x \\ & \\ &x=\frac{6}{4}=\frac{3}{2}=1.5 &&&& \text{ }x=\frac{-8}{4}=-2 \end{align}[/latex]

Figure 9

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

First we isolate the absolute value expression on one side of the equation.

[latex]\begin{align}1&=4|x - 2|+2 \\ -1&=4|x - 2| \\ -\frac{1}{4}&=|x - 2| \end{align}[/latex]

The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.

With both approaches, we will need to know first where the corresponding equality is true. In this case we first will find where [latex]|x - 5|=4[/latex]. We do this because the absolute value is a function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4. Solve [latex]|x - 5|=4[/latex].

[latex]\begin{align}x - 5&=4 & \text{ or } && {x - 5 }&={ -4 }\\ {x }&= {9} &\text{ or } && { x }&={ 1 } \end{align}[/latex]

After determining that the absolute value is equal to 4 at [latex]x=1[/latex] and [latex]x=9[/latex], we know the graph can change only from being less than 4 to greater than 4 at these values. This divides the number line up into three intervals:

[latex]{ x }<{ 1 },\text{ }{ 1 }<{ x }<{ 9 },\text{ and }{ x }>{ 9 }[/latex].

To determine when the function is less than 4, we could choose a value in each interval and see if the output is less than or greater than 4, as shown in the table below.

Because [latex]1\le x\le 9[/latex] is the only interval in which the output at the test value is less than 4, we can conclude that the solution to [latex]|x - 5|\le 4[/latex] is [latex]1\le x\le 9[/latex], or [latex]\left[1,9\right][/latex].

To use a graph, we can sketch the function [latex]f\left(x\right)=|x - 5|[/latex]. To help us see where the outputs are 4, the line [latex]g\left(x\right)=4[/latex] could also be sketched.

Figure 11. Graph to find the points satisfying an absolute value inequality.

We can see the following:

• The output values of the absolute value are equal to 4 at [latex]x=1[/latex] and [latex]x=9[/latex].
• The graph of [latex]f[/latex] is below the graph of [latex]g[/latex] on [latex]1<x<9[/latex]. This means the output values of [latex]f\left(x\right)[/latex] are less than the output values of [latex]g\left(x\right)[/latex].
• The absolute value is less than or equal to 4 between these two points, when [latex]1\le x\le 9[/latex]. In interval notation, this would be the interval [latex]\left[1,9\right][/latex].

We are trying to determine where [latex]f\left(x\right)<0[/latex], which is when [latex]-\frac{1}{2}|4x - 5|+3<0[/latex]. We begin by isolating the absolute value.

[latex]\begin{align}-\frac{1}{2}|4x - 5|&<-3 \hfill && \text{Multiply both sides by -2, and reverse the inequality}. \\ |4x - 5|&>6 \end{align}[/latex]

Next we solve for the equality [latex]|4x - 5|=6[/latex].

[latex]\begin{align}4x - 5&=6 & \text{or} && 4x - 5&=-6 \\ 4x - 5&=6 &&& 4x&=-1 \\ x&=\frac{11}{4} &&& x&=-\frac{1}{4} \end{align}[/latex]

Figure 12

Now, we can examine the graph of [latex]f[/latex] to observe where the output is negative. We will observe where the branches are below the x-axis. Notice that it is not even important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at [latex]x=-\frac{1}{4}[/latex] and [latex]x=\frac{11}{4}[/latex] and that the graph has been reflected vertically.

We observe that the graph of the function is below the x-axis left of [latex]x=-\frac{1}{4}[/latex] and right of [latex]x=\frac{11}{4}[/latex]. This means the function values are negative to the left of the first horizontal intercept at [latex]x=-\frac{1}{4}[/latex], and negative to the right of the second intercept at [latex]x=\frac{11}{4}[/latex]. This gives us the solution to the inequality.

[latex]x<-\frac{1}{4}\text{ }\text{or}\text{ }x>\frac{11}{4}[/latex]

In interval notation, this would be [latex]\left(-\infty ,-0.25\right)\cup \left(2.75,\infty \right)[/latex].