The number of prime numbers in the solution set of $| 2 x - 9 | + | 2 x - 39 | = 30$ is equal to

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Solution

$| 2 x - 9 | + | 2 x - 39 | = 30 = | (2 x - 9) - (2 x - 39) |$
The above equation is of the form $| a - b | = | a | + | b |$
So, $a b \leq 0$
i.e., $(2 x - 9) (2 x - 39) \leq 0$

$\Rightarrow x \in [\frac{9}{2}, \frac{39}{2}]$
$\Rightarrow x \in [4.5, 19.5]$
Prime numbers in the solution set are $5, 7, 11, 13, 17, 19$ .

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