Q48. - Algebra 2

Question

Hypothermia and hyperthermia are similar words but have opposite meanings. Hypothermia is defined as a lowered body temperature. Hyperthermia means extremely high body temperature, both conditions are potentially dangerous and occur when a person’s body temperature fluctuates by more than $\mathbf{8}\mathbf{°}$ from the normal body temperature of $\mathbf{98}\mathbf{.}\mathbf{6}{}^{\mathbf{\circ }}\mathbf{F}$. Write and solve an absolute value inequality to describe body temperatures that are considered potentially dangerous.

Step-by-Step Solution

Answer

The potentially dangerous body temperatures are$\mathbf{t}\ge \mathbf{106}\mathbf{.}\mathbf{6}{}^{\mathbf{\circ }}\mathbf{F}$ or $\mathbf{t}\le \mathbf{90}\mathbf{.}\mathbf{6}{}^{\mathbf{\circ }}\mathbf{F}$.

Step 1 – Description of step.

A normal body temperature of a person is $98.6{}^{\circ }F$. When person’s body temperature fluctuates by more than $8°$ from normal body temperature, the conditions hyperthermia and hyperthermia can occur.

 

Let $t$be the person’s body temperature.

Step 2 – Description of step.

Write an absolute value inequality to describe potentially dangerous body temperatures using the information in step 1.

 

$\left|t-98.6\right|\ge 8$

Step 3 – Description of step.

$\left|t-98.6\right|\ge 8$ is equivalent to $t-98.6\ge 8$ or $t-98.6\le -8$. Consider $t-98.6\ge 8$. Add $98.6$ on each side of the inequality.

$\begin{array}{l}t-98.6+98.6\ge 8+98.6\\ t\ge 106.6\end{array}$

Step 4 – Description of step.

Consider $t-98.6\le -8$. Add $98.6$ on each side of the inequality.

$\begin{array}{l}t-98.6+98.6\le -8+98.6\\ t\le 90.6\end{array}$

Therefore, the potentially dangerous body temperatures are $t\ge 106.6{}^{\circ }F$ or $t\le 90.6{}^{\circ }F$.