Question

Rozwiąż nierównosc

Answer 1

|x² - x| - |x - 5|  ≤  3

|x² - x| - |x - 5| - 3 ≤ 0

x² - x = 0     x(x-1) = 0   x = 0   x = 1

x - 5 = 0                       x = 5

x ∈ (-∞;0]∪(0;1]∪(1;5]∪(5;+∞)

1)x ∈ (-∞;0]
   +       -
|x²-x|-|x-5| - 3 = (x²-x) - (5 - x) - 3 = x² - 8
x² - 8 ≤ 0
x² ≤ 8
x ≤ 2√2
x ≥-2√2
x ∈ [-2√2; 0]

2)x ∈ (0;1]
   -       -
|x²-x|-|x-5| - 3 = -x² + x - (5 - x) -3 = -x² + 2x - 8
x² - 2x + 8  ≥ 0    ma rozwiązania !!!   zawsze > 0 (D < 0)
D = 4 -32 = -28 < 0
x ∈ (0;1]

3)x ∈ (1;5]
   +      -
|x²-x|-|x-5| - 3 = (x²-x) - (5 - x) - 3 = x² - 8
x² - 8 ≤ 0
x² ≤ 8
x ≤ 2√2
x ≥-2√2
x ∈ (1; 2√2]

4)x ∈ (5;+∞)
   +      +
|x²-x|-|x-5| - 3 = x² - x - x + 5 -3 = x² - 2x + 2
x² -2x + 2  ≤ 0 niemożliwo, nie ma rozwiązania !!! zawsze > 0 (D < 0)
D = 4 - 8 = -4
x ∈ (ø)

x ∈ (-∞; 0] ∪ (0; 1] ∪ (1;5] ∪ (5;+∞) => x ∈ [-2√2; 0] ∪ (0; 1] ∪ (1; 2√2] ∪ (ø) =>
=>x ∈ [-2√2; 2√2]