8.) /(3-2x) divided by (2+x)/ < 4
/ means absolute value
|(3-2x)/(2+x)|<4,
means:
-4 < (3-2x)/(2+x) < 4
Now its a good idea to sketch (3-2x)/(2+x), (see attachment)
Now solving (3-2x)/(2+x) = -4, gives x=-11/2 and so
-4 < (3-2x)/(2+x)
when x<-11/2, and alsowhen x>-2 (where the vertical asymtote is).
Now solve (3-2x)/(2+x) = 4, gives x=-5/6 and so:
(3-2x)/(2+x) < 4 for x>-5/6m and also when x<-2.
Combining these we see that both inequalities are satisfied when
x<-11/2 or x>-5/6.
RonL
Hi,
here I am again. You already know the solution of your problem, because CaptBlack has done it. I promised to come back with a solution. So here are my two cents:
abs((3-2x)/(2+x)) < 4 , x ≠ -2
First remove the absolute value:
A) (3-2x)/(2+x) > -4 and B) (3-2x)/(2+x) < 4
A) Multiply by the denominator. There are 2 possibilities:
A1) 3-2x > -4*(2+x) and x > -2
3 - 2x > -8 - 4x
11 > -2x
x < (-11)/2 and x > -2 that means: no solution
A2) 3-2x < -4*(2+x) and x < - 2
3 - 2x < -8 - 4x
11 < -2x
x < (-11)/2 and x < -2 that means: this set belongs to the solution
B) Multiply by the denominator. There are 2 possibilities:
B1) 3-2x < 4*(2+x) and x > -2
3 - 2x < 8 + 4x
-5 < 6x
x > (-5)/6 and x > -2 that means: this set belongs to the solution
B2) 3-2x < 4*(2+x) and x < -2
3 - 2x < 8 + 4x
-5 < 6x
x > (-5)/6 and x < -2 that means: no solution
tschüss
EB