Question 1210429
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1) Solve 2(√4x+5) - (√3x+1) = 6, where x is an integer.
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Tutor @greenestamps made a great step forward, guessing or finding mentally 
one integer solution  x = 5.


But when a solution to a Math problem,  or a math equation is found by the  " trial and error "
guessing method,  always a question arises - whether this solution is unique or whether other solutions exist.


There is one brilliant reasoning,  which may help in some cases,
and it will help to answer for this given equation.


Notice that in the left side of the equation the term  {{{sqrt(4x+1)}}}   grows  FASTER  than
the other term,  {{{sqrt(3x+1)}}}.       (This is obvious).


AFORTIORI,  the term  {{{2*sqrt(4x+1)}}}   grows even  FASTER  than the other term,  {{{sqrt(3x+1)}}}.


It means that the left side of the given equation is a monotonic function of  x
(monotonically increases for positive values of  x).


Hence,  this given equation,  having a constant value in its right side,  has a  UNIQUE  solution.


In other words - if it has one solution,  then this solution is unique.


Reasoning this way,  we transform our  " trial and error "  guessing method into  MATHEMATICALLY  STRICT   {{{highlight(highlight(proof))}}}.


So,  x = 5  is the unique real and integer solution to this given equation.


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                             Z Z Z  !


       Be careful !  That is true that in the positive 'x'-domain the left side monotonically increases.

       But what about negative 'x' ? - In this problem, the domain of negative 'x' is small and does not 
       make influence to the final conclusion !
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