# whats the absolute time interval?

For the time interval $1< \lvert t+1 \rvert \leq 3$ I am trying to solve for t to get my range to plot a function.

I know $\lvert t+1 \rvert \leq 3 \iff -3\leq t+1 \leq 3 \iff -4 \leq t \leq 2$

And then $\lvert t+1 \rvert > 1$ becomes $t+1 >1$ or $t+1 <-1$. Hence $t>0$ or $t<-2$.

Then things seem to contradict and i can't figure out my time range. What am I doing wrong?

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Your work is fine. You then need to find the $t$-values that satisfy both conditions: $-4 \leq t \leq 2$ and $t \in (-\infty, -2) \cup (0, \infty)$.
Alternatively, you could consider two cases: $t + 1 \geq 0$ and $t + 1 < 0$. When $t + 1 \geq 0$, we have that $|t + 1| = t + 1$, and thus $1 < t + 1 \leq 3$ so that $0 < t \leq 2$. All of these values are permissible as we are assuming $t > -1$ in this range as $t + 1 > 0$. You could deal with the other case $t + 1 < 0$ by noting that $|t + 1| = - (t + 1)$ in this range.