Bonjour

Résoudre dans R les inéquations suivantes.

x/x+2>1

x + 2 # 0

x # -2

x > 1(x + 2)

x > x + 2

x - x > 2

0 > 2 impossible donc pas de solution

-x/3x+1>-3

-x/(3x + 1) > -3(3x + 1)/(3x + 1)

-x/(3x + 1) + 3(3x + 1)/(3x + 1) > 0

(-x + 9x + 3)/(3x + 1) > 0

(8x + 3)/(3x + 1) > 0

8x + 3 = 0 et 3x + 1 # 0

8x = -3 et 3x # -1

x = -3/8 et x # -1/3

x................|-inf...........-3/8...........-1/3..........+inf

8x + 3.......|........(-).......o.....(+).............(+)....,......

3x + 1........|........(-)...............(-).......o.....(+)..........

Ineq..........|........(+)......o.....(-)........||......(+).........

x+2/x-1>x+1/x​

(x + 2)/(x - 1) - (x + 1)/x > 0

[x(x + 2) - (x + 1)(x - 1)]/[x(x + 1)(x - 1)] > 0

(x^2 + 2x - x^2 + 1)/[x(x + 1)(x- 1)] > 0

(2x + 1)/[x(x + 1)(x - 1)] > 0

2x + 1 = 0 et x # 0 et x - 1 # 0 et x + 1 # 0

2x = -1 et x # 0 et x # 1 et x # -1

x = -1/2 et x # 0 et x # 1 et x # -1

x.............|-inf........-1.......-1/2......0.......1........+inf

2x + 1.....|.......(-)........(-)....o...(+)....(+).....(+).........

x.............|.......(-).........(-).........(-).o..(+).....(+)........

x + 1.......|.......(-)....o...(+)......(+)......(+)......(+)........

x - 1........|.......(-).........(-)........(-)......(-)...o...(+).......

Ineq.......|.......(+)...||...(-)...o...(+).||...(-)..||....(+)......