11 a) x+3/12 x/12-3 x/9⇒x∈{1,3,9}
b)2x+1/15 2x/15-1 2x/14 x/7⇒ x∈{1,7}
c)2x-1/27 2x/27+1 2x/28 x/14⇒x∈{1,2,7,14}
d)x-5/6 x/6+5 x/11⇒ x∈{1,11}
12 a) x+1/x+6 x+1/x+1⇒ x+1/x+6-x-1 x+1/5 ⇒x+1∈{1,5} x+1=1 x=1-1=0 x+1=5 x=4
b)x+2/2x+29
x+2/x+2 x+2/2(x+2) x+2/2x+4 x+2/2x+29-2x-4 x+2/25⇒ x+2∈{1,5,25}
x+2=1 x=1-2 x=-1∉ N
x+2=5 x=5-2 x=3
x+2=25 x=25-2 x=23
c)2x-1/4x+19
2x-1/2x-1 2x-1/2(2x-1) 2x-1/4x-2 2x-1/4x+19-4x+2 2x-1/21 ⇒2x-1∈{1,3,7,21}
2x-1=1 2x=1+1 2x=2 x=1
2x-1=3 2x=3+1 2x=4 x=2
2x-1=7 2x=7+1 2x=8 x=4
2x-1=21 2x=21+1 2x=22 x=11
d)2x+1/7x+23 2x+1/2(7x+23) 2x+1/14x+26
2x+1/2x+1 2x+1/7(2x+1) 2x+1/14x+7 2x+1/14x+26-14x-7 2x+1/19⇒ 2x+1∈{1,19}
2x+1=1 2x=1-1 2x=0 x=0
2x+1=19 2x=19-1 2x=18 x=9
e)2x+5/7x+29 2x+5/2(7x+29) 2x+5/14x+58
2x+5/2x+5 2x+5/7(2x+5) 2x+5/14x+35 2x+5/14x+58-14x-35 2x+5/23⇒ 2x+5∈{1,23}
2x+5=1 2x=1-5 2x=-4 x=-2
2x+5=23 2x=23-5 2x=18 x=9
13 a) 3x+11=4x+15 3x-4x=15-11 -x=4 x=-4
a=3x+11 ⇔a=3*(-4)+11=-12+11=-1
b=4x+15 ⇔b=4*(-4)+15=-16+15=-1
a=-1 b=-1⇒ (a,b)=-1 [a si b sunt prime intre ele]
b)6x+13=4x+9 6x-4x=9-13 2x=-4 x=-2
a=6x+13 ⇔a=6*(-2)+13=-12+13=1
b=4x+9 ⇔b=4*(-2)+9=-8+9=1
a=1 b=1 ⇒(a,b)=1
