EJERCICIOS DE REEMPLAZO MARZO 5 DE 2010

1. 10 – 5x/2 ≥ 0

(20 – 5x)/2 ≥0

20 – 5x ≥ 2

-5x ≥ -18

X ≥ 18/5

X ≥ 3.6

INTERVALO DE SOLUCION: [3.6,∞)

5. 4 > (2 - 3x)/7 ≥ 2

28 > 2 – 3x ≥ 14

26 > -3x ≥ 12

-8.66 > x ≥ -4

PRUEBA: 4 > [2 – 3(-5)]/7 ≥ 2

28 > 2 + 15 ≥ 14

28 > 17 ≥ 14 = cumple la función

· 4 > [2 – 3(-3)]/7 ≥ 2

28 > 2 + 9 ≥ 14

28 > 11 ≥ 14 = no cumple la función

INTERVALO DE SOLUCION: (-8.66, -4]

6. x² ≥ -15x – 56

X² + 15x + 56 ≥ 0

a= 1

b= 15

c= 56

b² – 4ac

(15)² – 4(1)(56)

225 – 224

= 1 = 1> 0 = tiene dos soluciones

(-b ± √b²- 4ac)/2a

[-15 ± √(15)² – 4(1)(56)]/2

[-15 ± √1]/2

x₁ = (-15 + 1)/2 = -14/2 = -7

x₂ = (-15 – 1)/2 = -16/2 = -8

PRUEBA: (-6)² + 15(-6) + 56 ≥ 0

36 – 90 +56 ≥ 0

92 – 90 ≥ 0

2 ≥ 0

· (-9)² + 15(-9) + 56 ≥ 0

81 – 135 + 56 ≥ 0

137 – 135 ≥ 0

2 ≥ 0

INTERVALO DE SOLUCION: (-∞, -8] U [-7,∞)

10. -4/(1 + 2x) ≥ 0

-4 ≥ 1 + 2x

-2x ≥ 1 + 4

-2x ≥ 5

x ≥ -2.5

INTERVALO DE SOLUCION: [2.5,∞)

15. [(2x +1)² (x – 1)]/[x(x² – 1)] ≥ 0

[4x² + 4x + 1 (x – 1)]/ x[(x +1)(x – 1)] ≥ 0

(4x² + 4x + 1)/ (x² + x) ≥ 0

4x² + 4x + 1 ≥ x² + x

4x² – x² +4x –x + 1 ≥ 0

3x² + 3x + 1 ≥ 0

a= 3

b= 3

c= 1

b² - 4ac

(3)² – 4(3)(1)

9 – 12

= -3 = -3 < 0 = no tiene solución

18. [x(x² – 9)]/[(x² + x +1)(x – 1) ≥ 0

(x³ – 9x)/(x³ – x² + x² – x + x – 1) ≥ 0

(x³ – 9x)/(x³ – 1) ≥ 0

x³ – 9x ≥ x³ – 1

-9x ≥ -1

-9x + 1 ≥ 0

a= 0

b= -9

c= 1

b² – 4ac

81 – 0

= 81 = 81 > 0 = tiene dos soluciones

(-b ± √b²- 4ac)/2a

[9 ± √(-9)² – 4(0)(1)]/2(0)

(9 ± √81)/0

x₁= (9 + 9)/0 = ∞

x₂ = (9 – 9)/0 = 0

INTERVALO DE SOLUCION: [0, ∞)

20. (2 – x)/(x – 3) ≤ -2/x

x(2 – x) ≤ -2(x – 3)

2x – x² ≤ -2x + 6

-x² + 4x – 6 ≤ 0

a= -1

b= 4

c= -6

b² – 4ac

(4)² – 4(-1)(-6)

16 – 24

= -8 = -8 < 0 = no tiene solución

25. (5 + x²)/(2x + 4) ≤ 0

5 + x² ≤ 2x + 4

X² – 2x + 1 ≤ 0

a= 1

b= -2

c= 1

b² – 4ac

(-2)² – 4(1)(1)

4 – 4

= 0 = 0 tiene una solución

(-b ± √b²- 4ac)/2a

[2 ± √(2)² – 4(1)(1)]/2

(2± √0)/2

= 2/2 = 1

INTERVALO DE SOLUCION: (-∞, 1]

30. (x² – x + 3)/(x – 3) ≤ (18 + x²)/(x – 3)

(x – 3)(x² – x + 3) ≤ (18 + x²)(x – 3)

x² – x + 3 ≤ 18 + x²

-x ≤ 18 – 3

(-1)(-x ≤ 15)

x ≥ -15

INTERVALO DE SOLUCION: [-15, ∞)

35. [(4x - 4)/x] + [(1 + x)/x²] ≤ [4x/(1 + x)] + [4/(x² + x³)]

[4x – 4 + x(1 + x)]/x ≤ [4x + (x + x²)(4)]/x

(4x – 4 + x + x²)/x ≤ (4x + 4x + 4x²)/x

(x² + 5x – 4)/x ≤ (4x² + 8x)/x

x(x² +5x – 4) ≤ x(4x² + 8x)

x² + 5x – 4 ≤ 4x² + 8x

-4x² + x² + 5x - 8x – 4 ≤ 0

-3x² – 3x – 4 ≤ 0

a= -3

b= -3

c= -4

b² – 4ac

(-3)² – 4(-3)(-4)

9 – 48

= -39 = -39 < 0= no tiene solución