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Multiply by 2 (Posted on 2021-06-04) Difficulty: 3 of 5
Find all real solutions of

2x + 3x - 4x + 6x - 9x = 1

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution | Comment 1 of 4
By inspection x=0 is a solution.

Let f(x)=2x + 3x + 6x
g(x)= 4x + 9x
then the equation becomes f(x)-g(x)=1

These functions both have domain of the reals and range of the positive reals.  Also, both are strictly increasing over the reals.


Since f(-1)=1 there can be no solutions < -1
Since f(1)=11 and g(1)=13 there can be no solutions > 1

A quick look at the graph in this region shows there is indeed a single solution.  f(x)-g(x) reaches a maximum at (0,1).  This could easily be proven by finding the first and second derivatives if we needed to satisfy our Calc I teacher.

  Posted by Jer on 2021-06-05 10:27:40
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