## 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 “\begin{align*}t\end{align*}

\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.

\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*}

The horizontal line represents the variant of the normal body temperature. The intersection between the “\begin{align*}V\end{align*}

A few steps further: What does the point of the “\begin{align*}V\end{align*}

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*}

Our inequality would be \begin{align*}|t-98.6| \le 0.5\end{align*}

### Extension Investigation

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