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Inequalities Involving Modulus Function

The number of...

Question

The number of real solution of the equation $| a^{x} - 2 | + | 8 - a^{x} | < 5, a \in R$ is

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Solution

Given: $| a^{x} - 2 | + | 8 - a^{x} | < 5$
Assuming $a^{x} = t$
$\Rightarrow | t - 2 | + | 8 - t | < 5$
Minimum value of $| a | + | b | is | a + b |$
$\Rightarrow min (| t - 2 | + | 8 - t |) = | t - 2 + 8 - t | \Rightarrow min (| t - 2 | + | 8 - t |) = | 6 | > 5$

Hence, no solution exists.

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