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# Algebra

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Find all values of t such that floor(t) = 2t + 5. If you find more than one value, then list the values you find in increasing order, separated by commas.

Jun 18, 2021

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Write $t = n+x$ where $n$ is $\lfloor t\rfloor$ and $x$ is the fractional part ($0\le x<1$).  Given that $\lfloor t\rfloor = 2t+5$.  We have
\begin{eqnarray*}
\lfloor n+x \rfloor = 2(n+x)+5 \\
n = 2n+2x+5 \\
n = -5-2x
\end{eqnarray*}
Since $n$ is an integer, $2x$ must also be an integer.   And since $0\le x<1$,  we have $0\le 2x < 2$, so $x$ is either 0 or 0.5.  When $x=0$, we have $n=-5-2\cdot 0=-5$.  When $x=0.5$, we have $n=-5-2\cdot 0.5=-6$.  Thus, the possible values of $t$ in increasing order are $-6+.5=-5.5$, $-5+0=-5$.

Jun 18, 2021