lista00_cal1

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Lista de Exerc´ıcios 0
1. Resolva as inequac¸o˜es:
(a) 3x+ 3 < x+ 6
(b) x+ 6 ≤ 6x− 2
(c) x− 3 > 3x+ 1
(d) 2x > 3x
2. Estude o sinal:
(a) 3x+ 1
(b)
2− 3x
x+ 2
(c) (2x− 1)(x2 + 1)
(d)
2− x
3− x
3. Resolva as inequac¸o˜es:
(a) (x− 3)(x2 + 5) > 0
(b) x(x2 + 1) ≥ 0
(c) (2x+ 1)(x2 + x+ 1) ≤ 0
(d) x
x2+x+1
≥ 0
4. Verifique as identidades:
(a) x2 − a2 = (x− a)(x+ a);
(b) x3 − a3 = (x− a)(x2 + ax+ a2);
(c) xn − an = (x − a)(xn−1 + axn−2 +
. . .+ an−1), onde n 6= 0 e´ um natu-
ral.
5. Simplifique:
(a)
4x2 − 9
2x+ 3
(b)
1
x2
− 1
x− 1
(c)
(x+ h)2 − x2
h
(d)
x4 − p4
x− p
6. Fatore o polinoˆmio P (x)
(a) P (x) = x3 − 2x2 − x− 2
(b) P (x) = x4 − 3x2 + x2 + 3x− 2
(c) P (x) = x3 + 2x2 − 3x
(d) P (x) = x3 − 1
7. A afirmac¸a˜o: “quaisquer que sejam x e
y, x < y ⇔ x2 < y2”e´ falsa ou ver-
dadeira? Justifique.
8. Resolva as equac¸o˜es:
(a) |x+ 1| = 3
(b) |2x+ 3| = 0
(c) |2x− 1| = 1
(d) |x| = 2x+ 1
9. Resolva as inequac¸o˜es:
(a) |2x2 − 1| < 1
(b) |x+ 1| < |2x− 1|
(c) |x+ 3| > 1
(d) |x− 2|+ |x− 1| > 1
10. Elimine o mo´dulo:
(a) |x+ 1|+ |x|
(b) |2x− 1|+ |x− 2|
(c) |x− 2| − |x+ 1|
(d) |x|+ |x− 1|+ |x− 2|
11. Expresse o conjunto das soluc¸o˜es das in-
equac¸o˜es dadas em notac¸a˜o de interva-
los:
(a) x2 − 3x+ 2 < 0
(b) x2 + x+ 1 > 0
(c) x2 − 9 ≤ 0
(d) 2x−1
x+3
> 0
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