SOLUTION: Please please help me figure this out! I am tearing out my hair over this problem: x/x-3<=-8/x-6 I need to solve and state the solution set. Thank you !!!

Question 90877: Please please help me figure this out! I am tearing out my hair over this problem:
x/x-3<=-8/x-6 I need to solve and state the solution set. Thank you !!!
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) (Show Source): 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
------------
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.

Answer by jim_thompson5910(35256) (Show Source): 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

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