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# Graphs of Absolute Value Equations

## Graph two-variable absolute value equations

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Body Temperature
Teacher Contributed

## Real World Applications – Algebra I

### Topic

What’s normal for a person’s body temperature?

### Student Exploration

It’s common knowledge that a person’s normal body temperature is supposed to be 98.6 degrees. We can figure out using absolute value equations how much a person’s body temperature deviates from the norm for it to be considered abnormal (and possibly sick). Physicians say that people’s body temperature shouldn’t exceed 0.5 degrees from the norm. How can we represent this relationship as an absolute value equation, and then solve to know what the minimum and maximum body temperatures are?

Let’s say that “t\begin{align*}t\end{align*}” represents a person’s normal body temperature.

|t98.6|=0.5\begin{align*}|t-98.6|=0.5\end{align*}

This inequality means that the normal body temperature subtracted from the minimum and maximum body temperature should equal 0.5.

To solve this, we can break this up into two equations.

t98.6tt=0.5 and t98.6=0.5=98.6+0.5 and t=98.60.5=99.1 and t=98.1\begin{align*}t-98.6 &= 0.5 \ and \ t-98.6=-0.5\\ t &= 98.6+0.5 \ and \ t=98.6-0.5\\ t &= 99.1 \ and \ t = 98.1\end{align*}

This means that our normal body temperature should be between 98.1 and 99.1 degrees.

We can also graph this absolute value equation and see visually see what it means. Since we solved this equation, we can graph our solution set on a number line. We would also represent our solution space between the 98.1 and 99.1 tick marks on the number line. The solution space represents all of the different temperatures that are “normal” for humans.

We can also graph the solution space on an \begin{align*}xy\end{align*} coordinate graph, and interpret the solution. For this relationship, we’d have to graph two separate equations: \begin{align*}y=|x-98.6|\end{align*} and \begin{align*}y=0.5\end{align*} See below.

The horizontal line represents the variant of the normal body temperature. The intersection between the “\begin{align*}V\end{align*}” graph and the horizontal line is our solution 98. and 99.1.

A few steps further: What does the point of the “\begin{align*}V\end{align*}” graph represent on the graph? What do the \begin{align*}x\end{align*} values represent, in relation to body temperature? What do the \begin{align*}y\end{align*} values represent? If we were to shade the inside of the “\begin{align*}V\end{align*}” below the horizontal line, what would the solution space represent?

Let’s explore a little bit more deeply into this body temperature relationship and integrate absolute value inequalities in the equations and graphs. If we first had to integrate the use of an inequality sign instead of the equation \begin{align*}|t-98.6|=0.5\end{align*}, should we use a greater than sign, or a less than sign? Why?

Our inequality would be \begin{align*}|t-98.6| \le 0.5\end{align*} because the body temperature difference can’t be higher than 0.5 variance. If we were to represent this on a number line, we would have our solution space in between the endpoints at 98.1 and 99.1. This represents all of the temperatures that are considered “normal.”

### Extension Investigation

How else can you represent the maximum and minimum of something as an absolute value equation?

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