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This is a greatest integer equation. Solve for $x$: $$ [2x-7]=-3 $$

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  • $\begingroup$ $[2x-7]=[2x]-7$ $\endgroup$
    – obataku
    Sep 23, 2013 at 23:37

3 Answers 3

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We want to eliminate the floor; see wikipedia for some formulas. We then have $$-3\le 2x-7<-2$$ You can solve this for $x$ to get $$2\le x<2.5$$

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Hint: First, ignore the greatest integer sign and solve the equation. You will get a value for $x$. Then reflect on how much $x$ can increase before the left side gains a full unit.

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It's easy friend.
Firstly you should keep in mind that $x= [x] + \{x\}$ and $0 \leq \{x\}<1$
So, $2x-7 - \{2x-7\}=-3$
$\implies 2x-4 = \{2x-7\}$
$\implies 0 \leq (2x-4)<1$
$\implies 2 \leq x < 2.5 \implies x \in [2, 2.5)$
another equivalent method, $[2x -7] = [2x] -7 = -3 \implies [2x] = 4$
$\implies 2x - \{2x\} = 4 \implies 2x -4 = \{2x\}$ and you can solve as before.
In case, in the question $x \in \mathbb{N}$ then you get $x=2$.

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    $\begingroup$ Formatting tips here. $\endgroup$
    – Em.
    Apr 16, 2017 at 4:03

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