Esercizi prova intermedia Matematica

Risolvere le seguenti equazioni algebriche.

4 2

2

a − − ∨

2 3x + 7 = 4x + 11 x = x + 12 = 7x [x = 3 x = 4]

d

7

− −

3 x 2x 1 x 3 2

e x =0 [x = 0]

−

3 = + [x = 9]

b 2

6 5 15 √ √

h i

2 2

2 − − ∨

2 9x 2 = 0 x = x =

c f

3x + 7 = 0 [∅] 3 3

−

Luca Ferrari Corso di recupero di Matematica (1)

√ √

g p

2 4

− − ∨

x 6x + 4 = 0 x =3 5 x =3+ 5 x + 2 = 0 [∅]

√ √

q

3

8 4

2 − − − ∨

−3x ∨ − x 3x 4 = 0 x = 2 x = 2

5x = x = 0 x =

h 5

1

2 3 3

3 2

r

12x = 36x + 1 x =

i −9x−27 −3 ∨ − ∨

4x +12x = 0 x = x = x =

6 2 2

2

j 4 2

x + x + 5 = 0 [∅] s 3x + 7x + 2 = 0 [∅]

q

h i 8 2

− −

11

7 t (x 5) + (2x 1) + 3 = 0 [∅]

− 7

3x + 11 = 0 x =

k 3 2 2 4

u − −3]

(9 x ) + (x + 3) = 0 [x =

2 2 3

− −3 ∨ ∨

(x 9) = (x + 3) [x = x = 1 x = 6]

l 4 3 2

v −5x −13x −7x −1 ∨ ∨

x = 0 [x = x = 0 x = 7]

6 3

m − − −1 ∨

x 7x 8 = 0 [x = x = 2] 3 2 4

w 3x + 4x = 6x + 2x

q i

h

−5 12

n 5

x = 2 x = q q

h i

1 2 2

∨ ∨ − ∨

x =0 x = x = x =

q q i

h 2 3 3

12 12

6

o − − ∨

8x 1 = 0 x = x =

Risolvere le seguenti equazioni razionali fratte.

5 √ √

1 1 1 1 1 2

a −1 − ∨ −1

1+ = + x = 5 x = + 5 + + =0 [∅]

d 2

x +2 2 x 2x + 8 x x + 4x

7 2x + 5

2 −

x 4 e = [∅]

−2]

=0 [x =

b 2 −

9x + 54 x + 3x 18

−

x 2 −

1 x 1

x +1

2

2+ x x − −1 ∨

= [x = x = 1]

f

43

c − ∨

1= x = 0 x = 12

2 2

− −

−

2x x 1 4x

x

2

− −

2 x x 4x + 4

Risolvere le seguenti equazioni contenenti valori assoluti.

6 5

a e

|7 − |2 − −

2x| + 1 = 0 [∅] 3x + x| = 7 x x = 3

√

1

|4x −2 ∨ − − |x

+ 5| = 3 x = x =

b x|x + 1| 1 = + 1| x = 2

f

2 √ √

g

2

|x − − ∨

c |x |x − x| = x + 1 x =1 2 x =1+ 2

+ 2| + 3| = 0 [∅] 3

|2x − |x −2 ∨ |x| − |x − −

+ 1| + 5| = 0 [x = x = 4] 2| = 2x + 1 x =

d h 2

Risolvere le seguenti equazioni irrazionali. [Argomento non presente nel programma d’esame del corso di recupero]

7 √

√ √

x i

h 3

a − √

x 2x + 1 = 0 [x = 1] x +1+ =2 x =

d 2

x +1

√ √

h i √ √

3−1

2

−

2 x = x + 1 x =

b 3

e 2 − −1 ∨

x 1 = x + 1 [x = x = 3]

2

√ √ √

3

c 2

− − − − ∨

x 1 = 3 + x x [x = 2] 5 x =3 x [x = 1 x = 4]

f

Sistemi di equazioni

Risolvere i seguenti sistemi di equazioni a più incognite.

8 1

−

2x 3y = 13 x = 5 x + 2y + 1=0

a 4 [Impossibile]

f x 1

−1 −

3x + y = 14 y = = y

8 2

 

 

−

2x y + 2z = 11 x =2

−

3x y = 5  

[Indeterminato]

b g − −1

3x + y z = 2 y =

−6x −10

+ 2y =  

− −12

x + 2y 4z = z =3

 

−4

2x + 9y = x =7

c  

 

−

4x 3y + 2z = 18 x =2

− −2

3x = 19 y y =   −2

2x + 2y + 3z = 6 y =

h  

−

2x 3y = 5 + y − −14

3x + 6y 4z = z =2

 

[Impossibile]

d − −

7 x = x 4y 8

 

 

x =

2x + 3z = 2 5

  1

−

−14 −2 3x 2y + 2z = 3 y =

7x + y = x = i  

e 2 25

− −

−28 −

2x 2y 2z = 3

14x + 11y = y =0 z =

 

−

Luca Ferrari Corso di recupero di Matematica (2)

 38

   



− − −2 −2

x 2y + 3z = 1 x =

x + y 3z = x = 17

    75

2 2

m

j − −

∨

3x y + 2 = 0 x + z = 5 y =3 y =

[Indeterminato] 

 17

1

− −

2x 4y + 6z = 2 2x + y + z = 0 z =1 z =

    17

− −4

2x y + 7 = 0 x =0 x =

∨

k 2 2 − −1

x + y 4y = 21 y =7 y =  

 q q 

27 27

−

x = x =

2 2

x + y =4  

2 8 8

7 7

x 2 − n

x = x = ∨

+ y = 2 2 2  

q q

4 2

∨

7

l x + 9y = 9 5 5

± ±

5 1 y = y =

−

y = y =

−

3y = 2 x  

8 8

4 2

Disequazioni

Risolvere le seguenti disequazioni algebriche di primo grado.

9 a − − −1]

2x 3 < 3x 2 [x > 0 < 3x [x > 0]

f

g − −

25 3 > 2x 2(x 2) [∅]

− ≥ ≤ −

5 3x 2x + 7 x

b 32

− x > 0 [x < 0]

h

c −5x −1]

< 5 [x > − − − ∈

8x 2 > 3 8(1 x) [∀ x

i R]

7x > 9x [x < 0]

d − −

x 4 2 x 11 12

j − −

2 > 2x x >

e −6x −8]

48 < [x < 7

3 4 6

Risolvere le seguenti disequazioni algebriche di secondo grado.

10 2 2

a n

6 ≤ −3]

x > 0 [x = 0] x + 6x + 9 0 [x =

2 ≤

x 0 [x = 0]

b 3

5

2

o ∨ ≥ −

≥ ≤ − x

6x + 19x + 15 0 x 3 2

2

c −x ≤ ∈

0 [∀ x R] p 2 ≥ ∈

4x + 12x + 9 0 [∀ x R]

2

x < 0 [∅]

d 3

q

2

3 − −1 ∨

x < 2x 3 x < x > 2

2

e ∈

x +3 > 0 [∀ x R] 2

r ∈

x + x + 2 > 0 [∀ x R]

2 ≤

2x + 5 0 [∅]

f √ √ 2

s 8x > x + 16 [∅]

g

2 − ≥ ≤− ∨ ≥

x 7 0 x 7 x 7 √ √ √ i

h 4 4

5 5

2 3 3

2

− −

25 9x > 0 < x < − −

h t 3 4x > 0 < x <

3 3 2 2

32

2 − ∨

2x + 3x > 0 x < x > 0

i 3

2

u − ∨

2x > 3x + 9 x < x > 3

2

2

j ≤ ≤ ≤

x 5x [0 x 5] 2

v − − ∈

x 7x 2 < 0 [∀ x R]

2

−

4x x > 0 [0 < x < 4]

k 2

w ≥

x 2x + 3 [∅]

12

2 6

4x + 1 > 4x x =

l √ √

h i

1− 13 1+ 13

2

x

25

≥ − ≤ ≤

2 x x 3 x

m −5x − −1

3x + 2 > 0 < x < 2 2

Risolvere le seguenti disequazioni algebriche di grado superiore al secondo.

11 √ √

7 6

a ≤ ≤ − −

2x 0 [x 0] x 27 < 0 3 <x< 3

k

4

−3x 6

< 0 [x = 0]

b 4

− ≤

(x 3) 0 [x = 3]

l

6

c 5x < 0 [∅] 4 2

m −2 ∨

x > 2x + 8 [x < x > 2]

4

− ≥ ∈

(1 x) 0 [∀ x

d R] 6

n ≤

1 + 2x 0 [∅]

5

e −

3(8 x) > 0 [x < 8] 7

3 2

o − − −2 ∨

3x 7x 12x + 28 < 0 x < 2 < x <

√ √ 3

4 − − ∨

x 4 > 0 x< 2 x> 2

f p 6 3

√ x + 7x < 8 [−2 < x < 1]

g

5 5

−

2 x > 0 x< 2 q 4 ∈

x + 5 > 0 [∀ x R]

√ √

7 −6]

(x + 6) > 0 [x >

h

4 2

r − − ≤ − ≤ ≤

x 3x 10 0 5 x 5

q

h i

25

3 − 3

5x + 2 > 0 x >

i 12

3 2

s −2 ∨ −

2x + x < 8x + 4 x < < x < 2

q

h i 8 4 ∈

12 t x + 3x + 2 > 0 [∀ x

2 8

j − 6 ±

(1 2x ) > 0 x = R]

−

Luca Ferrari Corso di recupero di Matematica (3)

Risolvere le seguenti disequazioni razionali.

12 −

2x 5 3 5

−

(x + 1) 4(x + 1)

5

a ∨

< 0 x < x > 3 > 0

l

2

−

3 x 2

(x + 1)

1 2 43 12

3

∨ −1

−2 − ∨ −1 −

−

x >

+ > 0 < x < < x <

x <

b 2

x +1 x +2

1 1 2 2

(x + 1)(x + 3)(x + 4)

c > [0 < x < 2] m ≥ 0

x 2 3

−

3(5 x)

3 ∨

x + < 0 [x < 1 3 < x < 4]

d −4 ∨ −3 ≤

[x = x < 5]

−

x 4 6 5 2

e − − ≤

2x(2x + 1) x(2x + 1) (14 x ) 0 2 − −

− x 2x 8

7 x

n ≥ 6 −3]

+2 0 [x =

12

∨ ≤ ≤

−6 ≤ ≤ − 0 x 2

x 2

x +3 x + 6x + 9

4

− −

5 2x 9 x

73 73 52

o

≥ − ∨ ≤ ≥

0 x < < x 0

f 2 3 3

− − −

9x 49 (x 8)(1 x ) √ √

2 − −

x x 2

∨ ≤ ∨

≤− 3 1 <x 3 x> 2

x

g −3 ∨ −1 ∧ 6

> 0 [x < x > x = 2]

2 −

x + x 6

−

1 x 2x + 1 4 3 2

x + 3x + 2x

−8 ∨ −2 ∨

> [x < < x < 0 x > 2] p

h ≤ 0

−

x +2 2 x 7x + 3

3

x + 2x 3

∨

≤ −2 ∨ −1 ≤ − x = 0

x x <

> 0 [x > 0]

i 7

2

x + 3x + 7 3 5 1

2

(x + 2) q ≤

+

j ≥ −4 ∨ −2 ∨

0 [x < x = x > 4] 2 2 2

− − −

2x x x 4 x 4x + 4

2 −

x 16 √ √

2 2

− −2 ∨ − ≤ ≤ ∧ 6

(9 4x )(x + 7) < x < 0 6 2 6 x 6 + 2 6 x = 2

3 32

− ∨

> 0 < x < x > 3

k 2

−

3 x

Risolvere i seguenti sistemi di disequazioni.

13 2 2



≤ −

5x x x 3x + 10 < 0

c [∅]