SOLUTION: What happens if in a rational inequality the numerator is imaginary? This problem was on our test and now I need help for finals: Let f(x)={{{(x+3)/(x+1)}}} and g(x)={{{2/(x+4)}}}

Algebra ->  Rational-functions -> SOLUTION: What happens if in a rational inequality the numerator is imaginary? This problem was on our test and now I need help for finals: Let f(x)={{{(x+3)/(x+1)}}} and g(x)={{{2/(x+4)}}}      Log On


   

Question 306988: What happens if in a rational inequality the numerator is imaginary? This problem was on our test and now I need help for finals:
Let f(x)=%28x%2B3%29%2F%28x%2B1%29 and g(x)=2%2F%28x%2B4%29. For what values of x is the graph of y=f(x) below the graph of y=g(x)?
I didn't know exactly what to do, but I set f(x) less than g(x) and tried to solve for the solutions, but the numerator is imaginary and I don't know what to do.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Imaginary, what do you mean?
You can solve algebraically or by graphing.

f(x) is the red function.
g(x) is the green function.
You can clearly make out that between (-4,-1) that f(x) is less than g(x).
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Now let's show it algebraically.
Both function have singularities.
f(x) at x=-1 and g(x) at x=-4.
Break up the number line into three regions and check the inequality in those regions. You can pick any point in the region just not the end points.
1:(-infinity,-4)
2:(-4,-1)
3:(-1,infinity)
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Region 1: Let x=-5.
%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29
%28-5%2B3%29%2F%28-5%2B1%29%3C2%2F%28-5%2B4%29
%28-2%29%2F%28-4%29%3C-2
1%2F2%3C-2
No, not a valid region.
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Region 2: Let x=-2.
%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29
%28-2%2B3%29%2F%28-2%2B1%29%3C2%2F%28-2%2B4%29
%281%29%2F%28-1%29%3C2%2F%282%29
-1%3C1
Yes, a valid region.
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.
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Region 3: Let x=0.
%28x%2B3%29%2F%28x%2B1%29%3C2%2F%28x%2B4%29
%283%29%2F%281%29%3C2%2F%284%29
3%3C1%2F2
No, not a valid region.