TAREA 3

Desigualdades con denominador

2) Resolver las siguientes desigualdades, expresar la solución como intervalo y en la recta numérica.

a) 3/x≥3
Caso 1
x>0
3≥3x
3/3≥x
1≥ x
x≤1
x>0 ∩ x≤1
c.s. (0,1]
caso 2
x<0
3/x≤3
3≤3x
3/3≤x
x≥1
x<0>
c.s. (0)

c.s. (0) u (0, 1]

b) 5/x <>
Caso 1
x > 0
5 <>
5/1 / 6/7 <>
x > 35/6
x > 1 ∩ x > 35/6
c.s. (35/6, ∞)

caso 2
x <>
5 > 6/7 x
5/1 / 6/7 > x
x <>
x <>
c.s. (-∞, 0)

c.s. (-∞, 0) u (35/6, ∞)

c) x/ 2-4x ≤ 5/6
caso 1
2x - 4 > 0
2x > 4
x >2
x ≤ 5/6 (2-4x)
6(x ≤ 10/6- 20/6x
6x ≤ 10 – 20x
26x ≤ 10
x ≤ 10/26
x ≤5/13
x > 0 ∩ x ≤ 5/13
c.s. ( Ø )

Caso 2
x <>
x ≥ 5/6(2-4x)
6( x ≥ 10/6- 20/6x)
6x ≥ 10-20x
26x ≥ 10
x ≥ 10/26
x ≥ 5/13
x <>

c.s. [5/13, 2)

d) x/2x-3 > 5
caso 1
2x-3 >0
x > 3/2
x > 5(2x-3)
x > 10x-15
x-10x > -15
-1(-9x > -15)
9x <>
x <>
x <>
x > 3/2 ∩ x <>

c.s. (3/2, 5/3)

Caso2
2x-3<0
x<3/2
x<5(2x-3)
x<10x-15
-1(-9x<-15)
9x > 15
x> 5/3
x<3/2> 5/3
c.s. (Ø)

c.s. (Ø) u (3/2, 5/3)

e) -1<> -3-7x/4≤ 6
-3-4 <-7x ≤ 24-3
-1 (-7<-7x ≤ 21)
(7>7x ≥ -21)/ 7
1 >x ≥-3
c.s [-3, 1)