A Thermostat's Comfort Band

A smart thermostat lets the room drift within a few degrees of its setpoint before switching on heating or cooling. “Within d degrees of the setpoint” is an absolute value inequality, |T − s| ≤ d, and its edges, |T − s| = d, are where the system kicks in.

How it works

Absolute value equations and inequalities

In one sentence

For $d > 0$ , $|x - a| = d$ has the two solutions $x = a \pm d$ ; $|x - a| \le d$ is equivalent to $a - d \le x \le a + d$ ; and $|x - a| > d$ is equivalent to $x < a - d$ or $x > a + d$ .

General formula

|{\htmlData{sym=0}{T}} - {\htmlData{sym=1}{s}}| \le {\htmlData{sym=2}{d}} \iff {\htmlData{sym=1}{s}} - {\htmlData{sym=2}{d}} \le {\htmlData{sym=0}{T}} \le {\htmlData{sym=1}{s}} + {\htmlData{sym=2}{d}}, \qquad |{\htmlData{sym=0}{T}} - {\htmlData{sym=1}{s}}| = {\htmlData{sym=2}{d}} \iff {\htmlData{sym=0}{T}} = {\htmlData{sym=1}{s}} \pm {\htmlData{sym=2}{d}}

Absolute value measures distance from the setpoint. Inside the band the distance is small, and the band's two edges are the two temperatures exactly d away.

What the symbols mean

$T$
Room temperature
How warm the room is, in °C.
$s$
Setpoint
The temperature you asked for.
$d$
Allowed drift
How far the room may stray before the system switches on.

$|T - s|$ is the distance between the room temperature and the setpoint, whichever side it's on.
"Within $d$ degrees" means $|T - s| \le d$ , which unfolds into the double inequality $s - d \le T \le s + d$ : a band.
The band's edges solve $|T - s| = d$ , which has two answers: $T = s + d$ , where cooling starts, and $T = s - d$ , where heating starts.
Outside the band, $|T - s| > d$ splits into two separate pieces, too hot ( $T > s + d$ ) or too cold ( $T < s - d$ ), never both at once.

Learn the math

Absolute Value EquationsAlgebra · Lesson 19, Solving Equations and Inequalities Absolute Value InequalitiesAlgebra · Lesson 20, Solving Equations and Inequalities The Definition of a FunctionAlgebra · Lesson 24, Graphing and Functions