+-1098 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.
+7 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.
