Решите уравнение |x-|2x+1||=2.

1. Раскроем внешние знаки модуля:

|x - |2x + 1|| = 2; x - |2x + 1| = ±2; |2x + 1| = x ± 2.

2. Рассмотрим каждый случай:

a)

|2x + 1| = x - 2; {x - 2 ≥ 0;

{2x + 1 = ± (x - 2); {x ≥ 2;

{[2x + 1 = - (x - 2);

{[2x + 1 = (x - 2); {x ≥ 2;

{[2x + 1 = - x + 2;

{[2x + 1 = x - 2; {x ≥ 2;

{[2x + x = 2 - 1;

{[2x - x = - 2 - 1; {x ≥ 2;

{[3x = 1;

{[x = - 3; {x ≥ 2;

{[x = 1/3;

{[x = - 3; x ∈ ∅.

b)

|2x + 1| = x + 2; {x + 2 ≥ 0;

{2x + 1 = ± (x + 2); {x ≥ - 2;

{[2x + 1 = - (x + 2);

{[2x + 1 = (x + 2); {x ≥ - 2;

{[2x + 1 = - x - 2;

{[2x + 1 = x + 2; {x ≥ - 2;

{[2x + x = - 2 - 1;

{[2x - x = 2 - 1; {x ≥ - 2;

{[3x = - 3;

{[x = 1; {x ≥ - 2;

{[x = - 1;

{[x = 1; [x = - 1;

[x = 1.

Ответ: - 1; 1.