that would be (-infinity,-1) union (2,5)
this would be analyzed as follows, without graphing.
i also have a graph so you can see it visually.
the function is (x+1)/(x^2 - 7x + 10).
the denominator can be factored, so the function becomes (x+1)/((x-2)*(x-5)).
the function is 0 when x = -1.
the function have a vertical asymptote when x = 2 and when x = 5.
you would look at the following intervals.
x < 1
x > 1 and < 2
x > 2 and < 5
x > 5
you are looking for when the function is negative.
the interval will not include x = 1 because the function is 0 at that point.
the interval will not include 2 or 5 because the function is has a vertical asymptote at that point.
when x < 1, the function will be negative because you will have a negative numerator divided by a positive denominator.
it will be negative / (negative * negative) = negative / positive = negative.
when x is between -1 and 2, the function will be positive because you will have a positive numerator divided by a positive denominator.
it will be positive / (negative * negative) = positive / positive = positive.
when x is between 2 and 5, the function will be negative because you will have a positive numerator divided by a negative denominator.
it will be positive / (positive * negative) = positive / negative = negative.
when x is greater than 5, the function will be positive because you will have a positive numerator divided by a positive denominator.
it will be positive / (positive * positive) = positive / positive = positive.
some examples will show you what i mean.
for the interval between minus infinity and -1, pick any value.
i picked x = -2
the function becomes (-2+1)/((-2-2)*(-2-5)) = -1/(-4*-7) = -1/(28) = negative.
for the interval between -1 and 2, pick any value.
i picked x = 0
the function becomes (-+1)/((0-2)*(0-5)) = 1/(-2*-5) = 1/(10) = positive.
for the interval between 2 and 5, pick any value.
i picked 4.
the function becomes (4+1)/((4-2)*4-5)) = 5/(2*-1) = 5/-2 = negative.
for the interval between 5 and positive infinity, pick any value.
i picked 9.
the function becomes (9+1)/((9-2)*(9-5)) = 9/(7*4) = 9/28 = positive.
the following graphs shows that the analysis is correct.
the graph is negative when x is smaller than -1 and when x is between 2 and 5.
that's selection C.