Solve it Algebraically? I won't dare

Algebra Level 4

e − ∣ x ∣ + 1 = α \large e^{ -| x | }+1= \alpha

If the range of values of α \alpha for which the above equation has no solution can be expressed as ( − ∞ , A ] ∪ ( B , ∞ ) \left( -\infty ,A \right] \cup \left( B,\infty \right) , then find the value of A + B A+B .


The answer is 3.

Aditya Tiwari by Aditya Tiwari , 22, India

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4 solutions

Rohit Sachdeva
Apr 18, 2015

Graph of e − x e^{-x} lies between (0,1] for x>0

Hence graph of e − x + 1 e^{-x}+1 will lie between (1,2] for x>0 or |x|

Hence no soultion for α \alpha between ( − ∞ , 1 ] U ( 2 , ∞ ) (-∞,1]U(2,∞)

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Gautam Sharma
Apr 18, 2015

Firstly there needs to be a correction that is set values would be ( − ∞ , A ] ⋃ ( B , ∞ ) (-\infty ,A] \bigcup (B,\infty)

The range of the given function is ( 1 , 2 ] (1,2] Hence values of α \alpha would be ( − ∞ , 1 ] ⋃ ( 2 , ∞ ) (-\infty ,1] \bigcup (2,\infty)

Hence A+B=3

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Thanks for pointing out the mistake, Edited.

Aditya Tiwari - 6 years, 1 month ago

why it is given level 4 ? the sketch be seen here : https://www.desmos.com/calculator

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Atul Solanki
Apr 22, 2015

what an easy problemo

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