51 Résoudre les inéquations suivantes. 1. (x + 1)(x – 3) <(x + 1)(4x + 3) 2. x2 < x(-4x + 3) 3. x2 – 9> (x+3)(3x − 2) 4. 25x2 – 1 > (10x – 2)(2x +1) Bonjour pou

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]