Question

Let’s consider an absolute value in real life. For example, our internal body temperature has a range that is considered normal. Although many people think of the normal internal body temperature as 98.6, the temperature can vary by as much as plus or minus .5° and still be considered normal. This can be written as an absolute value function.

• Draw a number line and graph the normal internal body temperature. What is the minimum and maximum of our range? Label these on the number line.

• We can write this as an absolute value equation.  | x - 98.6 | ≤ 0.5
• Now apply what you have learned about absolute values to check if your equation is correct.

Answer:

|x - 98.6 |≤ 0.5 (the middle of the range is subtracted from the value of x and should be less than the range).

Check:

(x - 98.6) ≤ 0.5 and (x - 98.6) ≥ -0.5

x ≤ 98.6 + 0.5 and x ≥ 98.6 - 0.5

x ≤ 99.1 and x ≥ 98.1

This is the same as the graph above, so the equation is correct.

Assignment

Do ONE of these assignments:

1. The scenario depicted above is just one example of a real world use of an absolute value equation. Now find another real world example (it can be a similar range problem). Graph it on a number line, and then explain the minimum and maximum of the range. Write it as an absolute value and using what you have learned about solving absolute values, demonstrate that your equation is represented by the graph.
   OR
2. One of the most amazing compounds is water. It has three phases. What is the equation for the temperature of water as a liquid? This will take some thought. Think through the example that was shown above regarding the range of the normal internal body temperature.

Answer 1

Answer:-

Let us first consider the suggested example of Temperature of water.

We know Water has Three phases: Solid (Ice) , Liquid (Water) and Gas (Water Vapour) which depends on the temperature as given in following Number line.

Now , Let's define the Inequality for Liquid water:

We have for water to be liquid

Here , mid-point is and Range here is , Hence according to instructions given the Inequality is:

Now, Let us consider another Real world example:

The hearing range for Human Ear: Humans can hear Sounds having Frequency between 20 Hz and 20000 Hz.

Sound is categorized into 3 categories as per following Real line.

Now, let's define inequality for Frequency of sound (H) :