What is the soln of 2x+2|x|>2(2)1/2

Dear chilukuri

case 1 x>=0

2^x+2^|x|>2(2)^1/2

2^x+2^x>2(2)^1/2

2^x>(2)^1/2

x>=1/2

case 2   x<0

2^x+2^|x|>2(2)^1/2

or 2^x+2^-x>2(2)^1/2

or 2^2x+1>22^x(2)^1/2

simplify

 (2^x -2^1/2 -1)((2^x -2^1/2 +1)>0

first bracket is always negative so second bracket should also be negative

(2^x -2^1/2 +1)<=0

2^x <=2^1/2 -1

xln2 <=ln(2^1/2 -1)

x<=ln₂(2^1/2 -1)

so final solution

x<=ln₂(2^1/2 -1)         and x>=1/2

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