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 !!!

Algebra ->  Inequalities -> 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 !!!      Log On


   

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) About Me  (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) About Me  (Show Source):
You can put this solution on YOUR website!
x%2F%28x-3%29%3C=-8%2F%28x-6%29


Multiply both sides by the LCD


x%28x-6%29%3C=-8%28x-3%29 Simplify


x%5E2-6x%3C=-8x%2B24 Distribute

x%5E2%2B2x-24%3C=0 Get everything to one side

%28x%2B6%29%28x-4%29%3C=0 Factor the left side (note: if you need help with factoring, check out this solver)

So our zeros are x=-6 and x=4

This means we can plug in values near x=-6 and x=4 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 x=-7 (any point less than x=-6 will work)


f%28x%29=x%5E2%2B2x-24 Start with the given polynomial


f%28-7%29=%28-7%29%5E2%2B2%28-7%29-24 Plug in x=-7


f%28-7%29=%2849%29%2B2%28-7%29-24 Raise -7 to the second power to get 49


f%28-7%29=%2849%29%2B-14-24 Multiply 2 by -7 to get -14


f%28-7%29=11 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 f%285%29 (any point greater than x=4 will work)

Plug in x=5


f%28x%29=x%5E2%2B2x-24 Start with the given polynomial


f%285%29=%285%29%5E2%2B2%285%29-24 Plug in x=5


f%285%29=%2825%29%2B2%285%29-24 Raise 5 to the second power to get 25


f%285%29=%2825%29%2B10-24 Multiply 2 by 5 to get 10


f%285%29=11 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 x=-5 (any point in between x=-6 and x=4 will work)

Lets evaluate f%28-5%29

f%28x%29=x%5E2%2B2x-24 Start with the given polynomial


f%28-5%29=%28-5%29%5E2%2B2%28-5%29-24 Plug in x=-5


f%28-5%29=%2825%29%2B2%28-5%29-24 Raise -5 to the second power to get 25


f%28-5%29=%2825%29%2B-10-24 Multiply 2 by -5 to get -10


f%28-5%29=-9 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

-6%3C=x%3C=4


which looks like this in interval notation





If we graph y=x%5E2%2B2x-24, we can see the range that is below y=0

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2%2B2x-24%29+