I have the below absolute value inequalities word problem, I was able to get the answer but had to solve it by trying both <= and >= and then working out which one was correct. But the videos I see online for example, the authors seem to know the sign right away, without having to try both way. Could someone explain how to get the sign without working out the problem, also is it always <= as everything I see online it's always <=.
Here is the problem: The height of the plan must be within 2 inches of the standard 13-inch show size
|h - 13| ≤ 2
was the solution. Reading the problem, I know it's either 11 or 15.
When I solved it I tried both |height of the plan - 13| ≤ 2 and |height of the plan - 13| ≥ 2
worked them out to see the possible values to get ≤ as the correct sign, since the values can be 11 or 15.
You can break up one absolute value inequality into two simultaneous inequalities.
$|h-13|\le 2 \iff -2 \le h-13\le 2 \implies 11 \le h \le 15. $