2-7-67 inequality of rational function
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When looking at this problem, we're going to take everything
to one side and we're going to get a common denominator and
simplify.
So we get the equation -8 / X + 4 * X - 1 greater than 0.
To find the X intercepts, we set the numerator equal to 0, and
since there is no variable in the numerator, there aren't
going to be any X intercepts.
To find the Y intercepts, we put zero in for all the X's so -8
over 4 * -1 which would be two.
So 02 is a point.
The vertical asymptotes come from letting the denominator
equals 0, so X equal -4 and X equal 1.
Remember, we can never cross a vertical asymptote horizontal
asymptote.
The degree on bottom is bigger, so we're going to have it at y =
0.
Now we're going to draw in the graph.
We know this single .02 and we know we're never going to cross
the X axis because there were no X intercepts.
So we know that the graph is going to come down and go back
up in between those vertical asymptotes.
This vertical asymptote at X equal 1 came from this term
here, and it's to an odd power, odd multiplicity.
So if the YS on one side are positive, then the YS on the
other side have to be negative.
Can't cross the X axis because we don't have an X intercept.
We're going to get close to that horizontal asymptote as X goes
out to Infinity.
This other asymptote came from the X + 4, which is also to the
first power.
So when we look here, we know that whatever the X does on one
side, it's going to do the opposite on the other.
So if sorry the YS, if the Y is positive on one side, the Y on
the other side has to be negative.
We're not going to cross the X axis because we don't have any X
intercepts.
Now the problem is saying we want all of the values that are
greater than 0, IE we want all the Y values that are above the
X axis.
When we look at this graph, we can see that all the values that
are above the X axis are from -4 to one.
We're going to use parentheses there because it's above and we
can't equal at the asymptotes.