+743 Find all $x$ such that both $2x$ and $3x-7$ are in the interval $(-1,12)$.
+1804 We can write down two inequalities and then combine the values of x to get the interval.
The two inequalities we have are
\(-1 < 2x < 12 \space \space \space \space \space(1) \\ -1 < 3x - 7 < 12 \space \space \space \space \space (2) \)
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
\( 6 < 3x < 19 \\ 2 < x < 19 / 3\)
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! :)
