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Najlepsza Odpowiedź!
2010-03-25T08:14:46+01:00
A)
2 < I x I < 4 <=> I x I > 2 ∧ I x I < 4 <=>
<=> x ∈ (-∞, -2) u ( 2,+∞) ∧ x ∈ (-4 ; 4) <=>
<=> x ∈ (-4; -2) v x ∈ ( 2 ; 4)
b)
-3 < I x + 5 I ≤ 7 <=> x ∈ R ∧ I x +5 I ≤ 7 <=>
<=> x∈ R ∧ ( x+5 ≤ 7 ∧ x+5 ≥ -7 ) <=>
<=> x ∈ R ∧ ( x ≤ 7 - 5 = 2 ∧ x ≥ - 7 - 5 = -12 ) <=>
<=> x ∈ < -12 ; 2 >
c)
0 ≤ I x I < 5 <=> x ∈ R ∧ I x I < 5 <=>
<=> x ∈ R ∧ ( -5 < x < 5 ) < => x ∈ ( -5 ; 5)
d)
-2< I 2x - 6 I ≤ 3 < => x ∈ R ∧ ( -3 ≤ 2x - 6 ≤ 3 ) <=>
<=> x ∈ R ∧ ( 2x - 6 ≥ -3 ∧ 2x ≤ 3 +6 ) <=>
<=> x ∈ R ∧ ( 2x ≥ -3 + 6 = 3 ∧ 2x ≤ 9 ) <=>
<=> x ∈ R ∧ ( x ≥ 1,5 ∧ x ≤ 4,5 ) <=> x ∈ < 1,5 ; 4,5 >
e)
0 < I x - 2 I < 1 <=> x ∈ R \{2} ∧ I x - 2 I < 1 <=>
<=> x ∈ R \ { 2 } ∧ ( -1 < x -2 < 1 ) <=>
<=> x ∈ R \ { 2 } ∧ ( x - 2 > -1 ∧ x -2 < 1 ) <=.
<=> x ∈ R \{ 2 } ∧ ( x > -1 +2 = 1 ∧ x < 1 +2 = 3 ) <=>
<=> x ∈ R \ {2 } ∧ x ∈ (1 ; 3) <=> x ∈ (1 ;2) u (2; 3)
f)
4 ≤ I x - 1 I < 6 <=> I x - 1 I ≥ 4 ∧ I x - 1 I < 6 <=>
<=> ( - 4 ≥ x - 1 ≥ 4 ) ∧ ( -6 < x - 1 < 6 ) <=>
<=> ( x-1 ≤ -4 ∨ x -1 ≥ 4 ) ∧ ( x-1 >-6 ∧ x - 1 < 6 ) <=>
<=> ( x ≤ -4 +1 = -3 ∨ x ≥ 4+1 = 5 ) ∧ (x > -5 ∧ x < 7 ) <=>
<=> ( x ≤ -3 ∨ x ≥ 5 ) ∧ x ∈ ( -5 ; 7) <=>
<=> x∈ (-5; -3 > u < 5; 7)

2010-03-25T08:18:53+01:00
Rozwiąż nierówność:
a)2<|x|<4
|x|<4 i 2<|x|
-4<x<4 i x∈(-∞;-2)u(2;+∞)
odp. x∈(-4;-2)u(2;4)

b)-3<|x+5|≤7
|x+5|≤7 i -3<|x+5|
-7≤x+5≤7 i x∈R
-7-5≤x≤7-5 i x∈R
-12≤x≤2
odp. x∈<-12;2>

c)0≤|x|<5
|x|<5 i 0≤|x|
-5<x<5 i x∈R
odp.x∈(-5;5)

d)-2<2x-6|≤3
I2x-6|≤3 i -2<I2x-6|
-3≤2x-6≤3 i x∈R
-3+6≤2x≤3+6 i x∈R
3≤2x≤9 i x∈R
3/2≤x≤9/2
odp. x∈<1,5;4,5>

e)0<|x-2|<1
|x-2|<1 i 0<|x-2|
-1<x-2<1 i x∈R-{2}
-1+2<x<1+2 i x∈R-{2}
1<x<3 i x∈R-{2}
odp. x∈(1;2)u(2;3)

f)4≤|x-1|<6
|x-1|<6 i 4≤|x-1|
-6<x-1<6 i ( x-1≥4 lub x-1≤-4 )
-6+1<x<6+1 i ( x≥5 lub x≤-3 )
-5<x<7 i ( x≥5 lub x≤-3 )
odp. x∈(-5;-3>u<5;7)