Bonjour
Résoudre les inéquations suivantes.
1. (x + 1)(x – 3) <(x + 1)(4x + 3)
(x + 1)(x - 3) - (x + 1)(4x + 3) < 0
(x + 1)(x - 3 - 4x - 3) < 0
(x + 1)(-3x - 6) < 0
(x + 1) * 3(-x - 2) < 0
3(x + 1)(-x - 2) < 0
x + 1 = 0 et -x - 2 = 0
x = -1 et x = -2
x............|-inf.........-2........-1.........+inf
x + 1......|.......(-)..........(-)...o...(+).........
-x - 2....|.......(+).....o...(-).........(-)........
Ineq.....|.......(-)......o...(+)..o....(-)........
[tex]x \in ]-\infty ; -2[ U ]-1 ; +\infty[[/tex]
2. x2 < x(-4x + 3)
x^2 < -4x^2 + 3x
x^2 + 4x^2 - 3x < 0
5x^2 - 3x < 0
x(5x - 3) < 0
x = 0 et 5x - 3 = 0
x = 0 et 5x = 3
x = 0 et x = 3/5
x.............|-inf........0...........3/5..........+inf
x.............|.......(-)....o....(+)...........(+)............
5x - 3.....|.......(-)..........(-).....o....(+)..........
Ineq.......|.......(+)..o.....(-).....o.....(+).........
[tex]x \in ]0 ; 3/5[[/tex]
3. x2 – 9> (x+3)(3x − 2)
(x - 3)(x + 3) > (x + 3)(3x - 2)
(x - 3)(x + 3) - (x + 3)(3x - 2) > 0
(x + 3)(x - 3 - 3x + 2) > 0
(x + 3)(-2x - 1) > 0
x + 3 = 0 et -2x - 1 = 0
x = -3 et 2x = -1
x = -3 et x = -1/2
x.............|-inf........-3........-1/2.........+inf
x + 3......|.......(-).....o...(+)........(+)...........
-2x - 1....|.......(+)..........(+)...o....(-)...,.......
Ineq......|........(-).....o...(+)...o.....(-)..,,,,.....
[tex]x \in ]-3 ; -1/2[[/tex]
4. 25x2 – 1 > (10x – 2)(2x +1)
(5x - 1)(5x + 1) - (10x - 2)(2x + 1) > 0
(5x - 1)(5x + 1) - 2(5x - 1)(2x + 1) > 0
(5x - 1)(5x + 1 - 2(2x + 1)) > 0
(5x - 1)(5x + 1 - 4x - 2) > 0
(5x - 1)(x - 1) > 0
5x - 1 = 0 et x - 1 = 0
5x = 1 et x = 1
x = 1/5 et x = 1
x...............|-inf.........1/5.........1..........+inf
5x - 1........|.......(-)......o....(+)........(+).........
x - 1..........|........(-)............(-)...o....(+).......
Ineq.........|........(+)....o....(-)...o....(+).......
[tex]x \in ]-\infty ; 1/5[ U ]1 ; +\infty[[/tex]