Given f(x) = (x+2)/(x+4) and g(x) = (x+2)/(x-2) find when f(x)>=g(x)
I've plugged the reciprocals into Desmos, but I'm having difficulty
I'm also unsure of what to do with the interval chart.
when are they equal?
(x+2)/(x+4) = (x+2)/(x-2)
(x+2)(x-2) = (x+2)(x+4)
(x-2) =(x+4)
never
when does the denominator of f(x) = 0
when x = -4
if x < -4, like -5 , how do f and g compare?
f = -3/-1 = 3 and g = -3/-7 = 3/7
so when x<-4 f >g
now how about x between -4 and +2 when g blows up?
say x = 0
f = 2/4 and g = -1
so between x = -4 and +2, f>g
now what if x is > +2, like +3?
f = 5/7 and g = 5/1 Oh, now g>x
so I think f>g to the left of x = +2
(x+2)/(x+4) ≥ (x+2)/(x-2)
clearly x ≠ -4,2
We could look at the graph of f(x) and g(x) as a start
https://www.wolframalpha.com/input/?i=graph+y+%3D+%28x%2B2%29%2F%28x%2B4%29%2C+y+%3D+%28x%2B2%29%2F%28x-2%29
Wolfram graphed f(x) in blue and g(x) in red
notice where the blue curve is above the red curve, keep in mind that there are vertical asymptotes
at x = -4 and x = 2
from the graph it is clear that f(x) > g(x) for x < -4, then again for -2 < x < 2
let's see if if can find that algebraically:
(x+2)(x-2) = (x+4)(x+2)
x^2 - 4 = x^2 + 6x + 8
6x = -12
x = -2 , which is the x of our intersection
so we have 3 critical values: x = -4, x = -2 and x = 2
which means we have to investigate:
x < -4, -4 < x < -2, -2 < x < 2, and x > 2
test for x<-4 , let's pick -5
(x+2)/(x+4) ≥ (x+2)/(x-2)
LS = -3/-1 = 3
RS = -3/-3 = 1, and LS ≥ RS, so x < -4 works
test for -4 < x < -2 , let's pick x = -3
LS = -1/1 = -1
RS = -1/-5 = 1/5 , so not true, and -4<x<-2 is out
test for -2<x<2, let x = 0
LS = 2/4 = 1/2
RS = 2/-2 = -1, the statement is true, so -2 < x < 2 works
let's test for x > 2, let's pick x = 5
LS = 7/9
RS = 7/3 and since 7/9 < 7/3, x>2 is out
so we have :
x < -4 OR -2 < x < 2 , as seen in the graph
To find when f(x) is greater than or equal to g(x), we need to compare their values at different x-values. Let's first simplify the expressions for f(x) and g(x):
f(x) = (x+2)/(x+4)
g(x) = (x+2)/(x-2)
Now, let's find the common denominator for both expressions, which is (x+4)(x-2):
f(x) = (x+2)(x-2)/[(x+4)(x-2)]
= (x^2 - 4)/[(x+4)(x-2)]
g(x) = (x+2)(x+4)/[(x+4)(x-2)]
= (x^2 + 6x + 8)/[(x+4)(x-2)]
Now, we can compare f(x) and g(x) by finding their difference:
f(x) - g(x) = [(x^2 - 4)/[(x+4)(x-2)]] - [(x^2 + 6x + 8)/[(x+4)(x-2)]]
= (x^2 - 4 - x^2 - 6x - 8)/[(x+4)(x-2)]
= (-6x - 12)/[(x+4)(x-2)]
= -6(x + 2)/[(x+4)(x-2)]
Now, to determine when f(x) is greater than or equal to g(x), we need to consider the sign of -6(x + 2) and the sign of [(x+4)(x-2)].
1) Sign of -6(x + 2):
For -6(x + 2) to be non-negative (greater than or equal to zero), x + 2 must be non-positive (less than or equal to zero). Therefore, x ≤ -2.
2) Sign of [(x+4)(x-2)]:
To determine the sign of [(x+4)(x-2)], we can use an interval chart. An interval chart consists of three regions separated by the critical points (values that make the denominator zero), which in this case are x = -4 and x = 2.
Interval Chart:
_________________________
x | (x+4)(x-2)
-------------------------
x < -4 | (+)
-4 < x < 2 | (-)
x > 2 | (+)
From the interval chart, we can see that [(x+4)(x-2)] is positive (+) when x < -4 or x > 2, and it is negative (-) when -4 < x < 2.
Combining the information from both steps 1 and 2, we find that f(x) is greater than or equal to g(x) when x ≤ -2 and when -4 < x < 2.
Therefore, the solution to f(x) >= g(x) is x ≤ -2 or -4 < x < 2.