8. Ռացիոնալ անհավասարումների համակարգեր և համախմբեր

Վարժ․ 190

ա) { (x+1)(x-3)<0

{ (x+2)(x+1)<0 (1)

(x+1)(x-3)<0=> (x+1)(x-3)=0=> [ x+1=0; x-3=0=> [ x= -1; x=3

x ∈ (-1; 3)

(x+2)(x+1)<0=> (x+2)(x+1)=0=> [ x+2=0; x+1=0=> [ x= -2; x= -1

x ∈ (-2; -1)

(1)=> { x ∈ (-1; 3); x ∈ (-2; -1)

Պատ՝․ ⊘

բ) { x(x+5)<0

{ (x-1)(x-4)<0 (1)

x(x+5)<0=> x(x+5)=0=> [ x=0; x+5=0=> [ x=0; x= -5

x ∈ (-5; 0)

(x-1)(x-4)<0=> (x-1)(x-4)=0=> [ x-1=0; x-4=0=> [ x=1; x=4

x ∈ (1; 4)

(1)=> { x ∈ (-5; 0); x ∈ (1; 4)

Պատ՝․ ⊘

գ) { (x+2)(x-1)>0

{ (x+6)(x-3)<0 (1)

(x+2)(x-1)>0=> (x+2)(x-1)=0=> [ x+2=0; x-1=0=> [ x= -2; x=1

x ∈ (-∞; -2) ∪ (1; +∞)

(x+6)(x-3)<0=> (x+6)(x-3)=0=> [ x+6=0; x-3=0=> [ x= -6; x=3

x ∈ (-6; 3)

(1)=> { x ∈ (-∞; -2) ∪ (1; +∞); x ∈ (-6; 3)

Պատ՝․ ( -6; -2) ∪ (1; 3)

դ) { (x-5)(x-6)>0

{ (x+3)(x-4)<0 (1)

(x-5)(x-6)>0=> (x-5)(x-6)=0=> [ x-5=0; x-6=0=> [ x=5; x=6

x ∈ (-∞; 5) ∪ (6; +∞)

(x+3)(x-4)<0=> (x+3)(x-4)=0=> [ x+3=0; x-4=0=> [ x= -3; x=4

x ∈ (-3; 4)

(1)=> { x ∈ (-∞; 5) ∪ (6; +∞); x ∈ (-3; 4)

Վարժ․ 191

ա) { (x-1)(x-2)<0

{ x(x-3)>0 (1)

(x-1)(x-2)<0=> (x-1)(x-2)=0=> [ x-1=0; x-2=0=> [ x=1; x=2

x ∈ (1; 2)

x(x-3)>0=> x(x-3)=0=> [ x=0; x-3=0=> [ x=0; x=3

x ∈ (-∞; 0) ∪ (3; +∞)

(1)=> { x ∈ (1; 2); x ∈ (-∞; 0) ∪ (3; +∞)

Պատ՝․ ⊘

բ) { (x+10)(x-13)>0

{ (x+8)(x-12)<0 (1)

(x+10)(x-13)>0=> (x+10)(x-13)=0=> [ x+10=0; x-13=0=> [ x= -10; x=13

x ∈ (-∞; -10) ∪ (13; +∞)

(x+8)(x-12)<0=> (x+8)(x-12)=0=> [ x+8=0; x-12=0=> [ x= -8; x=12

x ∈ (-8; 12)

(1)=> { x ∈ (-∞; -10) ∪ (13; +∞); x ∈ (-8; 12)

Պատ՝․ ⊘

գ) { x²-4<0

{ x>9 (1)

x²-4<0=> (x-2)(x+2)=0=> [ x-2=0; x+2=0=> [ x=2; x= -2

x ∈ (-2; 2)

x>9=> x-9>0=> x-9=0=> x=9

x ∈ (9; +∞)

(1)=> { x ∈ (-2; 2); x ∈ (9; +∞)

Պատ՝․ ⊘

դ) { x<-2

{ x²-9>0 (1)

x<-2=> x+2=0=> x= -2

x ∈ (-∞; -2)

x²-9>0=> (x-3)(x+3)>0=> (x-3)(x+3)=0=> [ x-3=0; x+3=0=> [ x=3; x= -3

x ∈ (-∞; -3) ∪ (3; +∞)

(1)=> { x ∈ (-∞; -2); x ∈ (-∞; -3) ∪ (3; +∞)

Պատ՝․ (-∞; -3)

Վարժ․ 192

ա) { (x-1)(x-2)>0

{ (x-1)(x-3)>0 (1)

(x-1)(x-2)>0=> (x-1)(x-2)=0=> [ x-1=0; x-2=0=> [ x=1; x=2

x ∈ (-∞; 1) ∪ (2; +∞)

(x-1)(x-3)>0=> (x-1)(x-3)=0=> [ x-1=0; x-3=0=> [ x=1; x=3

x ∈ (-∞; 1) ∪ (3; +∞)

(1)=> { x ∈ (-∞; 1) ∪ (2; +∞); x ∈ (-∞; 1) ∪ (3; +∞)

Պատ՝․ (-∞; 1) ∪ (3; +∞)

բ) { (x+3)(x+2)<0

{ (x-4)(x+2)>0 (1)

(x+3)(x+2)<0=> (x+3)(x+2)=0=> [ x+3=0; x+2=0=> [ x= -3; x= -2

x ∈ (-3; -2)

(x-4)(x+2)>0=> (x-4)(x+2)=0=> [ x-4=0; x+2=0=> [ x=4; x= -2

x ∈ (-∞; -2) ∪ (4; +∞)

(1)=> { x ∈ (-3; -2); x ∈ (-∞; -2) ∪ (4; +∞)

Պատ՝․ (-3; -2)

գ) { (x+1)(x-1)>0

{ (x+1)(x-3)<0 (1)

(x+1)(x-1)>0=> (x+1)(x-1)=0=> [ x+1=0; x-1=0=> [ x= -1; x=1

x ∈ (-∞; -1) ∪ (1; +∞)

(x+1)(x-3)<0=> (x+1)(x-3)=0=> [ x+1=0; x-3=0=> [ x= -1; x=3

x ∈ (-1; 3)

(1)=> { x ∈ (-∞; -1) ∪ (1; +∞); x ∈ (-1; 3)

Պատ՝․ (1; 3)

դ) { (x+4)(x-6)>0

{ (x²-1)<0 (1)

(x+4)(x-6)>0=> (x+4)(x-6)=0=> [ x+4=0; x-6=0=> [ x= -4; x=6

x ∈ (-∞; -4) ∪ (6; +∞)

x²-1<0=> (x-1)(x+1)<0=> (x-1)(x+1)=0=> [ x-1=0; x+1=0=> [ x=1; x= -1

x ∈ (-1; 1)

(1)=> { x ∈ (-∞; -4) ∪ (6; +∞); x ∈ (-1; 1)

Պատ՝․ ⊘

Վարժ․ 197

ա) [ x²-49x+48>0

[ x²-x-6>0 (1)

x²-49x+48>0=> x²-49x+48=0=> [ x²-49x+48=0

x²-49x+48=0

D=b2-4ac=2209

x=49-47/2=2/2=1

x=96/2=48

x ∈ (-∞; 1) ∪ (48; +∞)

x²-x-6>0=> x²-x-6=0=> [ x²-x-6=0

x²-x-6=0

D=b²-4ac=1-(-24)=25

x=1-5/2= -4/2= -2

x=6/2=3

x ∈ (-∞; -2) ∪ (3; +∞)

(1)=> [ x ∈ (-∞; 1) ∪ (48; +∞); x ∈ (-∞; -2) ∪ (3; +∞)

Պատ՝․ (-∞; -1) ∪ (3; +∞)

բ) [ x²-4<0

[ 4/x>0 (1)

(x²-4)<0=> (x-2)(x+2)<0=> (x-2)(x+2)=0=> [ x-2=0; x+2=0=> [ x=2; x= -2

x ∈ (-2; 2)

4/x>0=> 4x>0=> 4x=0=> x=0

x ∈ (0; +∞)

(1)=> [ x ∈ (-2; 2); x ∈ (0; +∞)

Պատ՝․ (-2; +∞)

գ) [ 4x²-2x>0

[ 11x-4<0 (1)

4x²-2x>0=> x(4x-2)>0=> x(4x-2)=0=> [ x=0; 4x-2=0=> [ x=0; x=0,5

x ∈ (-∞; 0) ∪ (0.5; +∞)

11x-4<0=> 11x-4=0=> x=4/11

x ∈ (-∞; 4/11)

(1)=> [ x ∈ (-∞; 0) ∪ (0.5; +∞); x ∈ (-∞; 4/11)

Պատ՝․ (-∞; 4/11) ∪ (0.5; +∞)

դ) [ x²+2x-8>0

[ 5/x<12 (1)

x²+2x-8>0=> x²+2x-8=0

x²+2x-8=0

D=b²-4ac=4-(-32)=36

x= -2-6/2= -8/2= -4

x= -2+6=4/2=2

x ∈ (-∞; -4) ∪ (2; +∞)

5/x<12=> 5x-12<0=> 5x-12=0=> x=12/5

x ∈ (-∞; 12/5)

(1)=> [ x ∈ (-∞; -4) ∪ (2; +∞); x ∈ (-∞; 12/5)

Պատ՝․ x ∈ R (-∞; +∞)

ե) [ x²+1<0

[ x²+2x-24>0 (1)

x²+1<0=> x²+1=0=> ⊘

x²+2x-24>0=> x²+2x-24=0

x²+2x-24=0

D=b²-4ac=4-(-96)=100

x= -2-10/2= -12/2= -6

x=8/2=4

x ∈ (-∞; -6) ∪ (4; +∞)

(1)=> [ x ∈ ⊘; x ∈ (-∞; -6) ∪ (4; +∞)

Պատ՝․ (-∞; -6) ∪ (4; +∞)

զ) [ x²-9>0

[ x²+x-2<0 (1)

x²-9>0=> x²-9=0=> (x-3)(x+3)=0=> [ x-3=0; x+3=0=> [ x=3; x= -3

x ∈ (-∞; -3) ∪ (3; +∞)

x²+x-2<0=> x²+x-2=0

x²+x-2=0

D=b²-4ac=1-(-8)=9

x= -1-3/2= -4/2= -2

x= -1+3/2= 2/2=1

x ∈ (-2; 1)

(1)=> [ x ∈ (-∞; -3) ∪ (3; +∞); x ∈ (-2; 1)

Պատ՝․ (-∞; -3) ∪ (-2; 1) ∪ (3; +∞)