Question

|||x - 2| + 1| - x| = 5 - x​

Answer 1

первый случай

||x−2+1|−x|=5−x, x>=2

||x−1|−x|=5−x, x>=2

|x−1−x|=5−x, x>=2

|−1|=5−x, x>=2

1=5−x, x>=2

x=4, x>=2

x=4 - корень

второй случай

||-x+2+1|−x|=5−x, x<2

||-x+3|−x|=5−x, x<2

|-x+3−x|=5−x, x<2

|-2x+3|=5−x, x<2

первый подслучай

-2x+3=5−x, x<2, -2x+3>=0

-x=2, x<2, x<=1.5

x=-2, x<=1.5 - корень

второй подслучай

2x-3=5−x, x<2, -2x+3<=0

3x=8, x<2, x>=1.5

x=8/3, x<2, x>=1.5 - между прочим, не корень

Answer 2

To solve the equation ||x - 2| + 1| - |x|| = 5 - x, we need to consider different cases based on the absolute value expressions.

Case 1: x ≥ 2 In this case, the absolute values simplify as follows: |x - 2| = x - 2 |1| = 1 |x| = x

Substituting these values back into the equation, we have: (x - 2) + 1 - x = 5 - x

Simplifying further: x - 2 + 1 - x = 5 - x -1 = 5 - x

Rearranging the equation: x = 5 + 1 x = 6

However, we need to check if x = 6 satisfies the original equation. Substituting x = 6 into the equation: ||6 - 2| + 1| - |6|| = 5 - 6 |4 + 1| - 6 = -1 |5| - 6 = -1 5 - 6 = -1 -1 = -1

The equation is satisfied for x = 6 in this case.

Case 2: x < 2 In this case, the absolute values simplify differently: |x - 2| = -(x - 2) = -x + 2 |1| = 1 |x| = -x

Substituting these values back into the equation, we have: (-x + 2) + 1 - (-x) = 5 - x

Simplifying further:

• x + 2 + 1 + x = 5 - x 3 = 5 - x

Rearranging the equation: x = 5 - 3 x = 2

However, we need to check if x = 2 satisfies the original equation. Substituting x = 2 into the equation: ||2 - 2| + 1| - |2|| = 5 - 2 |0 + 1| - 2 = 3 |1| - 2 = 3 1 - 2 = 3 -1 = 3

The equation is not satisfied for x = 2 in this case.

Therefore, the solution to the equation ||x - 2| + 1| - |x|| = 5 - x is x = 6.