Answer:
(7/3,3)
Step-by-step explanation:
first part is -3 < 2/(3x-9).
second part is 2/(3x-9) < -1.
there is a third piece that needs to be evaluated, which is where the vertical asymptote lies.
the vertical asymptote lies at x = 3.
when x < 3, the graph of 2/(3x-9) is negative and stays negative.
when x > 3, the graph of 2/(3x-9) is positive and stays positive.
this is clear from the equation, because when x > 3, (3x-9) is positive, and 2 is positive and therefore the fraction is positive.
this is also clear from the equation, because when x < 3, (3x-9) is negative, and therefore 2/(3x-9) is negative because the numerator is positive and the denominator is positive and therefore the fraction is negative.
what this tells you is that 2/(3x-9) is greater than -3 when x is greater than 3.
what this also tells you is that 2/(3x-9) is not smaller than -1 when x is greater than 3.
if you solve the equations for when -3 = 2/(3x-9) and when -1 = (2/(3x-9), you will find that:
-3 = 2/(3x-9) when x = 25/9 which is equal to 2.778 rounded to 3 decimal places.
-1 = 2/(3x-9) when x = 7/3 which is equal to 2.333 rounded to 3 decimal places.
at x = 25/9, 2/(3x-9) = -3
when x < 25/9, 2/(3x-9) > -3.
when x > 25/9, 2/(3x-9) < -3.
however, when x > 3, 2/(3x-9) becomes positive and is therefore greater than -3 all over again.
therefore 2/(3x-9) is greater than -3 when x is smaller than 25/9 and when x is greater than 3.
this satisfies the inequality of -3 < 2/(3x-9).
the other inequality that needs to be satisfied is 2/(3x-9) < -1.
you know that 2/(3x-9) = -1 when x = 7/3.
if you look at the interval between x = 7/3 and x = 3, you will see that 3/(3x-9) is less than -1.
for example, when x = 8/3, which is greater than 7/3 and less than 3, 2/(3x-9 is equal to -30 which is clearly less than -1.
when x is less than 7/3, 2/(3x-9) is greater than -1.
for example, when x is equal to 6/3, which is less than 7/3, 2/(3x-9) is equal to -2/3 which is greater than -1.
therefore when x > 7/3 and < 3, y = 2/(3x-9) is less than -1.
what you really want to do is graph the equality points where the graph shows equality, and then look at the intervals between those points.
the following graph shows you what i mean.
i graphed 4 equations.
first equation is y = 2/(3x-9)
second equation is y = -3
third equation is y = -1
fourth equation is x = 3.
you can see from the graph that y = -3 is smaller than y = 2/(3x-9) when x is smaller than 25/9 and when x is greater than 3.
you can also see from the graph that y = (3x-9) is smaller than y = -1x is greater than 7/3 and when x is smaller than 3.
those are your solutions.
-3 < 2/(3x-9) when x < 25/9 or when x > 3.
in interval notation, this would be x = (minus infinity, 25/9) union (3, plus infinity).
