Theory of Equations 1 Question 47
48. Find the set of all solutions of the equation
$$ 2^{|y|}-\left|2^{y-1}-1\right|=2^{y-1}+1 $$
(1997 C, 3M)
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Solution:
- Given, $2^{|y|}-\left|2^{y-1}-1\right|=2^{y-1}+1$
Case I When $y \in(-\infty, 0]$
$$ \begin{array}{lc} \therefore & 2^{-y}+\left(2^{y-1}-1\right)=2^{y-1}+1 \\ \Rightarrow & 2^{-y}=2 \\ \Rightarrow & y=-1 \in(-\infty, 0] \end{array} $$
Case II When $y \in(0,1]$
$$ \begin{aligned} & \therefore \quad 2^{y}+\left(2^{y-1}-1\right)=2^{y-1}+1 \\ & \Rightarrow \quad 2^{y}=2 \\ & \Rightarrow \quad y=1 \in(0,1] \end{aligned} $$
Case III When $y \in(1, \infty)$
$$ \begin{array}{rlrl} & \therefore & 2^{y}-2^{y-1}+1 & =2^{y-1}+1 \\ \Rightarrow & 2^{y}-2 \cdot 2^{y-1} & =0 \\ \Rightarrow & 2^{y}-2^{y} & =0 \text { true for all } y>1 \end{array} $$
From Eqs. (i), (ii) and (iii), we get
$$ y \in{-1} \cup[1, \infty) $$
