Question

Как решить двойное неравенство с модулем?

Answer 1

двойное неравенство (любое, хоть с модулем, хоть без модуля)))
равносильно системе неравенств !!
К примеру:1<|3+X|<2
{ |3+X|<2
{ 1<|3+X|
-----------система
{ -2 < 3+X < 2 (здесь еще одна система, т.е. пересечение неравенств)
{ 3+X > 1   или   3+X < -1 (здесь объединение неравенств)
-----------------------------------система
т.е. решаем каждое неравенство уже без модуля и
ищем пересечение промежутков...
{ -5 < X < -1
{ X > -2   или   X < -4
-----------------------------система
Решение: -5 < X < -4   или   -2 < X < -1

Answer 2

Solving Double Inequality with Absolute Value

To solve a double inequality with absolute value, you can follow these steps:

1. Isolate the Absolute Value Expression: - First, isolate the absolute value expression on one side of the inequality.

2. Split the Inequality: - Then, split the inequality into two separate inequalities, one for when the expression inside the absolute value is positive and one for when it's negative.

3. Solve Each Inequality: - Solve each inequality separately to find the range of values that satisfy the original double inequality.

For example, if you have an inequality like |ax + b| < c, you would split it into two separate inequalities: ax + b < c and -(ax + b) < c, and then solve each one individually.

This method allows you to find the range of values for the variable that satisfy the original double inequality.