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Find all values of t such that \lfloor t\rfloor = 3t + 4 - \lfloor 2t \rfloor.  If you find more than one value, then list the values you find in increasing order, separated by commas.

 Dec 31, 2023
 #1
avatar+6 
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Let’s solve this step by step:

First, let’s consider the case where

t

is an integer. In this case,

⌊t⌋=t

and

⌊2t⌋=2t

. Substituting these into the equation gives

t=3t+4−2t

, which simplifies to

t=−4

. However,

−4

is not a solution because when we substitute

−4

into the original equation, we get

−4=−8+4−(−8)

.

Now, let’s consider the case where

t

is not an integer. In this case,

t

can be written as

t=n+r

where

n

is an integer and

0≤r<1

. Substituting this into the equation gives

n=3n+3r+4−2n−2r

, which simplifies to

n=−4

and

r=0

. However,

r=0

implies that

t

is an integer, which contradicts our assumption that

t

is not an integer.

Therefore, there are no values of

t

that satisfy the given equation. The equation

⌊t⌋=3t+4−⌊2t⌋

has no solutions.

 Dec 31, 2023

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