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Algebra Level 4

$\large \dfrac { { \left( \left\lfloor x \right\rfloor -1 \right) \left( \left\lfloor x \right\rfloor -2 \right) }^{ 2 }\left( \left\lfloor x \right\rfloor +4 \right) }{ \left( \left\lfloor x \right\rfloor +2 \right) \left( \left\lfloor x \right\rfloor -3 \right) } <0$

Find the number of integral values that satisfy the inequality above.

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Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

3 1 4 5 2

1 solution

Sam Bealing
May 21, 2016

As the question asks about integral values we can just ignore the floor functions as the floor of an integer is equal to itself:

If $x=2$ then the expression is equal to $0$ so $x \neq 2$ :

$\dfrac{(x-2)^2 (x-1) (x+4)}{(x-3) (x+2)}<0 \Rightarrow \dfrac{(x-1) (x+4)}{(x-3) (x+2)}<0 \: (as \: (x-2)^2>0)$

The above expression just becomes a matter of case checking resulting in:

$-4<x<-2 \lor 1<x<2 \lor 2<x<3$

As the inequalites are strict, the only value satisfying the equations is $x=-3$ giving the answer as:

$\boxed{1}$

Whoa! Wait? What?

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