disequazioni_2
1
Disequazioni con valore assoluto e trascendenti A. Disequazioni con valori assoluti 1. |x| +2x ≤ 3 Sol. (-∞, 1] 2. 1 -|x +1| > 2x +1 Sol. (-∞, - 1 3 ) 3. |x +1|- 1 ≤ x 2 Sol. (-∞, 0] ∪ [1, +∞) 4. 1+ x + x 2 < |x| Sol. ∅ 5. 1 -|x 2 - 1|≥ x Sol. [-2, 0] ∪{1} 6. 1+ x ≤|1+ x + x 2 | Sol. (-∞, +∞) 7. |x +1|- 1 ≥|x| Sol. [0, +∞) 8. |x - 1| + |x +1|≥ 3x +2 Sol. (-∞, 0] 9. |x| + |x 2 - 1| > 1 Sol. (-∞, -1) ∪ (-1, 0) ∪ (0, 1) ∪ (1, +∞) 10. |x - 1|≥|x 2 - 3x +2| Sol. [1, 3] 11. |x 2 - x +1| < |x 2 - 1| Sol. (0, 1 2 ) ∪ (2, +∞) 12. |1 - 2x + x 2 |≥|x + x 2 | Sol. (-∞, 1 3 ] 13. 1 |x| ≥ x Sol. (-∞, 0) ∪ (0, √ 5-1 2 ] 14. x |x-1| ≥ x - 1 Sol. (-∞, 1) ∪ (1, 2+ √ 2] 15. x+1 x-1 ≥ 1 Sol. [0, 1) ∪ (1, +∞) 16. x x-2 ≤ x Sol. [0, 1] ∪ [3, +∞) B. Disequazioni esponenziali 1. 2 x ≥ 3 Sol. [log 2 3, +∞) 2. 3 x+1 < 5 Sol. (-∞, log 3 5 - 1) 3. 4 x/3 > 2 Sol. ( 3 2 , +∞) 4. 5 1-2x < 1 Sol. ( 1 2 , +∞) 5. 2 x-1 +2 x +2 x+1 > 1 Sol. (log 2 ( 2 7 ) , +∞) 6. 3 x-1 +3 x +3 x+1 - 3 ≥ 0 Sol. [log 3 ( 9 13 ) , +∞) 7. 3 x 2 -2x+1 ≤ 1 Sol. {1} 8. 2 1-x 2 > 8 Sol. ∅ 9. ( 1 2 ) 3-x 2 ≤ 1 Sol. (-∞, - √ 3) ∪ [ √ 3, +∞) 10. 3 2x > 9 · 3 x Sol. (2, +∞) 11. 4 2x-1 > 2 2x+4 Sol. (3, +∞) 12. 8 2-x - 4 1+x ≤ 0 Sol. [ 4 5 , +∞) 13. 3 x - 5 · 2 -x ≥ 0 Sol. [log 6 5, +∞) 14. 3 2x - 3 1+x - 18 ≤ 0 Sol. (-∞, log 3 6] 15. 2 2x - 3 · 2 x +2 ≤ 0 Sol. [0, 1] 16. 3 2x - 2 · 3 x - 2 > 1 Sol. (1, +∞) 17. 2 2x -3 3x 4 4x -5 5x ≥ 0 Sol. (-∞, 0) ∪ (0, +∞) 18. 2 x -2 -x 2 x +2 -x < 1 Sol. (-∞, +∞) 19. 2 x 3 x -2 x > 1 Sol. (0, 1 log 2 3-1 ) 20. 2 2x -2 x+1 2 x -1 ≤ 0 Sol. (0, 1] 21. 2 x+1 x ≥ 1 Sol. (-∞, -1] ∪ (0, +∞) 22. 2 x 2 x - √ 2-2 x > 0 Sol. (0, 1] C. Disequazioni logaritmiche 1. log 3 x ≤ 3 Sol. (0, 27] 2. 1 + log 2 x ≥ 1 2 Sol. [ 1 √ 2 , +∞) 3. log 1/2 (x + 1) ≤ 1 Sol. [- 1 2 , +∞) 4. log 2 (1 + x 2 ) > 2 Sol. ( √ 3, +∞) 5. log 10 x> log 10 (x - 1) Sol. ∅ 6. log 3 (x 2 - 2x) ≤ 0 Sol. (0, 1+ √ 5 2 ] 7. log 2 (log 3 x) > 0 Sol. (3, +∞) 8. log 2 (1 + log 2 x) > 1 Sol. (2, +∞) 9. log 10 (log 2 ( 1 x ) ) > 0 Sol. (0, 1 2 ) 10. log 2 (x - x 2 ) - log 2 (1 - x) > 1 Sol. ∅ 11. log 2 10 x - 3 log 10 x +2 > 0 Sol. (0, 10) ∪ (100, +∞) 12. log 2 2 x - 5 log 2 x +6 ≤ 0 Sol. [4, 8] 13. log 3 (1 - √ x) ≤-1 Sol. [ 4 9 , 1) 14. log 2 (x - √ 1 - x 2 ) < 0 Sol. ( 1 √ 2 , 1) 15. 1-log 2 x log 2 (1-x) < 0 Sol. (0, 1) 16. log 2 x log 2 2 x-1 < 0 Sol. (0, 1 2 ) ∪ (1, 2)
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Transcript of disequazioni_2
Disequazioni con valore assoluto e trascendenti
A. Disequazioni con valori assoluti
1. |x| + 2x ≤ 3 Sol. (−∞, 1] 2. 1 − |x + 1| > 2x + 1 Sol. (−∞,− 13 )
3. |x + 1| − 1 ≤ x2 Sol. (−∞, 0] ∪ [1, +∞) 4. 1 + x + x2 < |x| Sol. ∅5. 1 − |x2 − 1| ≥ x Sol. [−2, 0] ∪ {1} 6. 1 + x ≤ |1 + x + x
2| Sol. (−∞, +∞)
7. |x + 1| − 1 ≥ |x| Sol. [0, +∞) 8. |x − 1| + |x + 1| ≥ 3x + 2 Sol. (−∞, 0]
9. |x| + |x2 − 1| > 1 Sol. (−∞,−1) ∪ (−1, 0) ∪ (0, 1) ∪ (1, +∞)
10. |x − 1| ≥ |x2 − 3x + 2| Sol. [1, 3]
11. |x2 − x + 1| < |x2 − 1| Sol. (0,12 ) ∪ (2, +∞) 12. |1 − 2x + x2| ≥ |x + x2| Sol. (−∞,
13 ]
13.1|x| ≥ x Sol. (−∞, 0) ∪ (0,
√5−12 ] 14.
x|x−1| ≥ x − 1 Sol. (−∞, 1) ∪ (1, 2 +
√2]
15.
∣
∣
∣
x+1x−1
∣
∣
∣≥ 1 Sol. [0, 1) ∪ (1, +∞) 16.
∣
∣
∣
xx−2
∣
∣
∣≤ x Sol. [0, 1] ∪ [3, +∞)
B. Disequazioni esponenziali
1. 2x ≥ 3 Sol. [log2 3, +∞) 2. 3x+1 < 5 Sol. (−∞, log3 5 − 1)
3. 4x/3 > 2 Sol. (32 , +∞) 4. 51−2x < 1 Sol. (1
2 , +∞)
5. 2x−1 + 2x + 2x+1 > 1 Sol. (log2
(
27
)
, +∞) 6. 3x−1 + 3x + 3x+1 − 3 ≥ 0 Sol. [log3
(
913
)
, +∞)
7. 3x2−2x+1 ≤ 1 Sol. {1} 8. 21−x2
> 8 Sol. ∅9.
(
12
)3−x2
≤ 1 Sol. (−∞,−√
3) ∪ [√
3, +∞) 10. 32x > 9 · 3x Sol. (2, +∞)
11. 42x−1> 22x+4
Sol. (3, +∞) 12. 82−x − 41+x ≤ 0 Sol. [45 , +∞)
13. 3x − 5 · 2−x ≥ 0 Sol. [log6 5, +∞) 14. 32x − 31+x − 18 ≤ 0 Sol. (−∞, log3 6]
15. 22x − 3 · 2x + 2 ≤ 0 Sol. [0, 1] 16. 32x − 2 · 3x − 2 > 1 Sol. (1, +∞)
17.22x−33x
44x−55x≥ 0 Sol. (−∞, 0) ∪ (0, +∞) 18.
2x−2−x
2x+2−x< 1 Sol. (−∞, +∞)
19.2x
3x−2x> 1 Sol. (0,
1log2 3−1 ) 20.
22x−2x+1
2x−1 ≤ 0 Sol. (0, 1]
21. 2x+1
x ≥ 1 Sol. (−∞,−1] ∪ (0, +∞) 22.2x
2x−√
2−2x> 0 Sol. (0, 1]
C. Disequazioni logaritmiche
1. log3 x ≤ 3 Sol. (0, 27] 2. 1 + log2 x ≥ 12 Sol. [ 1√
2, +∞)
3. log1/2(x + 1) ≤ 1 Sol. [− 12 , +∞) 4. log2(1 + x2) > 2 Sol. (
√3, +∞)
5. log10 x > log10(x − 1) Sol. ∅ 6. log3(x2 − 2x) ≤ 0 Sol. (0,
1+√
52 ]
7. log2(log3 x) > 0 Sol. (3, +∞) 8. log2(1 + log2 x) > 1 Sol. (2, +∞)
9. log10(log2
(
1x
)
) > 0 Sol. (0,12 ) 10. log2(x − x2) − log2(1 − x) > 1 Sol. ∅
11. log210 x − 3 log10 x + 2 > 0 Sol. (0, 10) ∪ (100, +∞) 12. log2
2 x − 5 log2 x + 6 ≤ 0 Sol. [4, 8]
13. log3(1 −√x) ≤ −1 Sol. [ 49 , 1) 14. log2(x −
√1 − x2) < 0 Sol. ( 1√
2, 1)
15.1−log2 xlog2(1−x) < 0 Sol. (0, 1) 16.
log2 xlog2
2x−1
< 0 Sol. (0,12 ) ∪ (1, 2)
