You can
put this solution on YOUR website!x/(x-3) <= -8/(x-6)
----------------------
1st: Find values x cannot take:
x=3 and x=6
2nd: Plot those on a number line:
Comment: That breakes the number line into 3 intervals.
Check each interval with a test value to see where the
solution set is.
3rd: Test point check.
Interval (-inf,-3)
Test Point x=-10
Substitute in the original inequality:
-10/(-13)<=-8/-16; 10/13 <= 1/2 ; False ; therefore no solutions in (-inf,-3)
----------------
Interval (-3,6)
Test Point x=0
Substitute in the original inequality x/(x-3) <= -8/(x-6):
0 <= -8/-6; 0<=4/3; True ; therefore interval (-3,6) is part of the solution
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Inerval (6,+inf)
Test Point x=10
Substitute in the original inequality:
10/7 <=-8/4 10/7 <- -2; False; therefore no solutions in (6,+inf)
=====================
Final Answer: -3 < x < 6 is the solution set.
=================
Cheers,
Stan H.
You can
put this solution on YOUR website!
Multiply both sides by the LCD
Simplify
Distribute
Get everything to one side
Factor the left side (note: if you need help with factoring, check out this
solver)
So our zeros are
and
This means we can plug in values near
and
to find out if they produce y values that are less than zero:
Now lets test everything from negative infinity to x=-6
Plug in
(any point less than x=-6 will work)
Start with the given polynomial
Plug in
Raise -7 to the second power to get 49
Multiply 2 by -7 to get -14
Now combine like terms
So x=-7 produces a y value greater than zero
So everything less than x=-6 will produce a y value greater than zero. So lets ignore this range
---------------------------
Now lets test everything from x=4 to infinity
Lets evaluate
(any point greater than x=4 will work)
Plug in
Start with the given polynomial
Plug in
Raise 5 to the second power to get 25
Multiply 2 by 5 to get 10
Now combine like terms
So everything greater than x=4 will produce a y value greater than zero. So lets ignore this range.
------------------------------------
Now lets test everything from x=-6 to x=4
Plug in
(any point in between x=-6 and x=4 will work)
Lets evaluate
Start with the given polynomial
Plug in
Raise -5 to the second power to get 25
Multiply 2 by -5 to get -10
Now combine like terms
So everything from x=-6 to x=4 will produce a y value less than zero
So the solution set is
which looks like this in interval notation
If we graph
, we can see the range that is below y=0
(