96.8°f ≤ 0.4 ≤ 102.2°f

Question

Write and solve an absolute value inequality that represents the range of possible temperatures for a reading of 99.1°f . Explain what you need to consider and calculate. Than graph the solution set

Answer 1

Assuming that 99.1℉ is a temperature rounded to 1 decimal place then the true temperature T must satisfy T<99.15 and T>99.05, that is, 99.05<T<99.15 (ignoring the traditionally accepted rule for rounding T≥99.05).

To express this as an absolute value inequality we subtract 99.1 from each term in the multiple inequality:

-0.05<T-99.1<0.05. This is now in the form ready to be converted to an absolute value inequality:

|T-99.1|<0.05. The range of values of T making this true is shown graphically below.

Continued in comment...

 

Comments

The blue section of the line represents the inequality, excluding the orange points and line. What we’ve done here is to consider what’s meant by “absolute value”: the distance between zero and two points equidistant from zero. Then we adjust the zero point—T-99.1=0 so that we have the two points 99.05 and 99.15 equidistant from this zero point. This gives us the blue range of values for T that produce a reading of 99.1℉ accurate to one decimal place.

If we now consider the temperature range 96.8 and 102.2 we take the average=(96.8+102.2)/2=99.5℉, and we subtract this from each of the other temperatures and we get -2.7<T-99.5<2.7, which becomes |T-99.5|<2.7. The number line graph can be altered so that the zero point is 99.5 and the end points are 96.8 and 102.2.