let S be the set of all real numbers except 1 Define the bin

let S be the set of all real numbers except -1. Define the binary operation * on S by a*b=a+b+ab 1) show that is a group 2) solve 5*x*2*3=6 in S

2> 5*x*2*3 =6

=>(5+x+5x)*(2+3+2*3)=6

=> (5+6x)*11=6

=> 5+6x+11+11(5+6x)=6

=> 16+6x+55+66x=6

=> 72x +71=6

=> 72x= 6-71

=> 72x=-65

=> x= -65/72 (answer)

1> addition and multiplication of real numbers will also be a real number , So closure property holds in S

Let a,b,c belong to S

a*(b*c)= a*(b+c+bc) = a+b+c+bc+a(b+c+bc)= a+b+c+ab+ac+bc+abc

(a*b)*c= (a+b+ab)*c = a+b+c+ab+(a+b+ab)c= a+b+c+ab+ac+bc+abc

so (a*b)*c=a*(b*c) thus S is associative

Let e be an element in S such that a*e=e*a =a for all a belongs to S

a*e=a

=> a+e+ae =a

=> e+ae=0

=> e(e+1)=0

=> e= 0 since e can\'t be -1

so 0 is the identity element of S

let m belong to S such that a*m=m*a= 0

a*m=0

=> a+m+am=0

=> m+am = -a

=>m(1+a)=-a

=> m = -a/(1+a)

so , m will always exists in S as a can\'t be -1

thus each element of S has the inverse in S

Hence, (S,*) is a group