Video Transcript
If 𝑥 is greater than negative six
but less than or equal to four, then what interval does three 𝑥 minus five lie
in?
In this question, we’re told the
value of 𝑥 is between negative six and four. 𝑥 can be equal to four, but it
can’t be equal to negative six. We then need to find the interval
in which the expression three 𝑥 minus five lies, which is essentially the reverse
process of solving an inequality. Let’s consider what operations have
been performed to 𝑥 to give the expression three 𝑥 minus five. Well, firstly, 𝑥 has been
multiplied by three and then we’ve subtracted five. We can do the same to the endpoints
of the interval in which 𝑥 lies.
Firstly, multiplying each part of
the inequality by three, we find that three 𝑥 is greater than negative 18 but less
than or equal to 12. And then subtracting five from each
part, we find that three 𝑥 minus five is greater than negative 23 and less than or
equal to seven. We can therefore express this as an
interval from negative 23 to seven. At the lower end of the interval,
we have a strict inequality sign. Negative 23 is not included in the
interval. So we use an open bracket. At the upper end of the interval,
we have a weak inequality sign. Seven is included in the interval,
so the interval is closed at this end.
And so, we’ve completed the
problem. Three 𝑥 minus five lies in the
interval from negative 23 to seven, which is open at the lower end and closed at the
upper end.
