Question 134556
if {{{a}}} is the number, then the additive inverse is {{{-a}}} and its reciprocal is {{{1/a}}}.  "is the same as" means "equals"


The additive inverse of a number ({{{-a}}}), divided by 12 ({{{-a/12}}}) is the same as (=) one less than ({{{-1}}}) three times its reciprocal ({{{3*(1/a)=3/a}}}).  Put it all together:


{{{-a/12=(3/a)-1}}}


Add 1 to both sides:
{{{1-a/12=3/a}}}


LCD on the left is 12, so:
{{{(12-a)/12=3/a}}}


Cross-multiply (numerator on the left times denominator on the right equals denominator on the left times numerator on the right)
{{{a(12-a)=36}}}


{{{12a-a^2=36}}}
{{{-a^2+12a-36=0}}}


Multiply by -1:
{{{a^2-12a+36=0}}}


Trinomial is a perfect square:
{{{(a-6)(a-6)=0}}}


Hence {{{a=6}}}


Check:
{{{-(6)/12=(3/6)-1}}}
{{{-1/2=1/2-1=-1/2}}} Answer checks.