Find all $x$ such that both $2x$ and $3x-7$ are in the interval $(-1,12)$.
We can write down two inequalities and then combine the values of x to get the interval.
The two inequalities we have are
−1<2x<12 (1)−1<3x−7<12 (2)
Now, let's isolate x from each equation to find the values of x.
From the first equation, we get that
−1/2<x<6
From the second equation, we get that
6<3x<192<x<19/3
Since we must take the most restrictive interval, the overlapping values of both gets us the interval
2<x<6 or (2,6) in interval notation .
Thanks! :)