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If y=3[x]+1=4...

Question

If $y = 3 [x] + 1 = 4 [x - 1] - 10,$ then the value of $[x + 2 y] =$ , , where $[.]$ denotes GIF.

107.0

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107

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Solution

Given: $y = 3 [x] + 1 = 4 [x - 1] - 10$
Using GIF property, $[x - 1] = [x] - 1$
$\Rightarrow 3 [x] + 1 = 4 [x] - 4 - 10$
$\Rightarrow [x] = 15$
$\Rightarrow y = 3 [x] + 1 = 3 \times 15 + 1 = 46$
$\Rightarrow 2 y = 46 \times 2 = 92$

We know $[x] \leq x < [x] + 1$
$\Rightarrow 15 \leq x < 15 + 1$
$\Rightarrow 15 + 92 \leq x + 2 y < 16 + 92$
$\Rightarrow 107 \leq x + 2 y < 108$
Also, $[x + 2 y] \leq x + 2 y < [x + 2 y] + 1$

Comparing the above two relations we get, $[x + 2 y] = 107$

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