⌊ 5 − ⌊ x ⌋ ⌋ = 1 5
If all the values of x that satisfy the equation above are in the interval a ≤ x < b , find the product a b .
Note:
⌊
x
⌋
is the floor function, or the greatest integer function.
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amzng........
very well said!
Let y = ⌊ x ⌋
As
y
is an integer so
⌊
5
−
y
⌋
=
5
−
y
=
1
0
⇒
y
=
−
1
0
For any − 1 0 ≤ x < − 9 , ⌊ x ⌋ = − 1 0
So here ( a = − 1 0 , b = − 9 ) a b = ( − 1 0 ) ( − 9 ) = 9 0
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In order for the equation to be true, 5 − ⌊ x ⌋ must be greater than or equal to 1 5 and less than but not equal to 1 6 i.e.
1 5 ≤ 5 − ⌊ x ⌋ < 1 6
1 0 ≤ − ⌊ x ⌋ < 1 1
− 1 1 < ⌊ x ⌋ ≤ − 1 0 .
Since ⌊ x ⌋ is an integer, ⌊ x ⌋ = − 1 0 , as this is the only integer in the interval ( − 1 1 , − 1 0 ] .
x must therefore be greater than or equal to − 1 0 and less than but not equal to − 9 i.e. − 1 0 ≤ x < − 9 .
a = − 1 0 , b = − 9 , a b = 9 0 .