Question

Rezolvati in Z ecuatia : |2x-5| + |10-4x| =9 dau funda va rog am nevoie

Answer 1

Varianta I:

|2x-5| + |10-4x| = 9

|2x-5| + |4x-10| = 9

|2x-5| + |2•(2x-5)| = 9

|2x-5| + 2•|2x-5| = 9

3•|2x-5| = 9

|2x-5| = 9:3

|2x-5| = 3

2x-5 = 3 ∨ 2x-5 = -3

2x = 8 ∨ 2x = 2

x = 4 ∨ x = 1

=> x ∈ {1; 4}

Varianta II:

|2x-5| + |10-4x| = 9

|2x-5| = 9 - |10-4x|

2x-5 = -9 + |10-4x| ∨ 2x-5 = 9 - |10-4x|

|10-4x| = 2x+4 ∨ |10-4x| = 14-2x

10-4x = -2x-4 ∨ 10-4x = 2x+4

∨ 10-4x = -14+2x ∨ 10-4x = 14-2x

2x = 14 ∨ 6x = 6 ∨ 6x = 24 ∨ 2x = -4

x = 7 ∨ x = 1 ∨ x = 4 ∨ x = -2

Observăm că x = 7 și x = -2 nu verifică.

=> x ∈ {1; 4}

Answer 2

| a | = a, daca a>0

| a | = -a, daca a<0

| a | = 0, daca a=0

Facem tabelul de semne pentru 2x-5 si 10-4x

x         | -oo               5/2           +oo

---------------------------------------------------

2x-5   |  - - - - - - - - - - 0 + + + + + + +

--------------------------------------------------

10-4x  | + + + + + + + + 0 - - - - - - - - -

2x-5=0 => 2x=5 => x=5/2

10-4x=0 => 10=4x => x=10/4=5/2

Cazul I: x∈(-oo; 5/2)

|2x-5|= -(2x-5)= -2x+5

( 2x-5<0 in cazul asta)

|10-4x|=10-4x

(10-4x>0 in cazul asta)

|2x-5| + |10-4x|= 5-2x+10-4x=15-6x

15-6x=9 => 6x=15-9 => 6x=6 => x=1 ∈ (-oo; 5/2) si e intreg, deci e solutie

Cazu II: x=5/2

Atunci  |2x-5| + |10-4x| = |0| + |0| = 0 + 0=0, si ecuatia |2x-5| + |10-4x|=9 nu are solutie reala

Cazul III: x∈(5/2; +oo)

2x-5>0 => |2x-5| = 2x-5

10-4x<0 => |10-4x|= -(10-4x)= -10+4x

|2x-5| + |10-4x|=2x-5 -10+4x=6x-15

6x-15=9 => 6x=9+15 => 6x=24 => x=24/6=4 ∈ (5/2; +oo), e intreg, deci e solutie

S={1, 4}