solve-x-R-2-x-1-3-4-x-2-4-

Question Number 175333 by mnjuly1970 last updated on 27/Aug/22

s o l v e (x \in R) .

⌊ \frac{2^{x} + 1}{3} ⌋ + ⌊ 4^{\frac{x}{2}} ⌋ = 4

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Answered by mr W last updated on 27/Aug/22

t = \frac{2^{x} + 1}{3}

⌊ t ⌋ + ⌊ 3 t - 1 ⌋ = 4

t = n + f

n + ⌊ 3 n + 3 f - 1 ⌋ = 4

n + 3 n - 1 + ⌊ 3 f ⌋ = 4

5 - 4 n = ⌊ 3 f ⌋ ⩾ 0 \Rightarrow 4 n ⩽ 5 \Rightarrow n ⩽ 1

5 - 4 n = ⌊ 3 f ⌋ < 3 \Rightarrow 4 n > 2 \Rightarrow n ⩾ 1

\Rightarrow n = 1

⌊ 3 f ⌋ = 5 - 4 n = 1

\Rightarrow 1 ⩽ 3 f < 2

\Rightarrow \frac{1}{3} ⩽ f < \frac{2}{3}

\Rightarrow 1 + \frac{1}{3} ⩽ t < 1 + \frac{2}{3}

\Rightarrow \frac{4}{3} ⩽ \frac{2^{x} + 1}{3} < \frac{5}{3}

\Rightarrow 4 ⩽ 2^{x} + 1 < 5

\Rightarrow 3 ⩽ 2^{x} < 4

\Rightarrow \log_{2} 3 ⩽ x < 2

Commented by mnjuly1970 last updated on 27/Aug/22

t h a n k z a l o t s i r W

Commented by Tawa11 last updated on 28/Aug/22

Great sir

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