Help with math question please!?

Question

So it needs to be solved. 1/(x+1) + 3/(x-4) >(or equal to)
-1/2. The answer to the question is (- infinity, -5] u (1,2} u (4, infinity). I don't understand how to do it, a math lab person told me the 2 critical values are 1 and 4, but I don't think thats all there is. thank you so much for your help!

Thank you very much for your help, i understand that there are regions, then you are meant to test points, but how do i get the (1,2] from the middle of the answer? (with inequalities you can't multiply/divide with terms using X because you don;t know the sign, which may change the inequality direction)

Answer 1

If you work it out as given then part of the answer is (-INF, -6] . However if you change (x+1) to (x-1) then the answer you give is true. So I have done that.

(x - 1) = 0 if x = 1 and 1/0 is not good
(x - 4) = 0 if x = 4 and the same as the above
This means that x can not have one of these values.

Now lets look at the equality part of the stated relation. You can also do this with the inequality in place but you have to be careful since we are dealing with fractional terms and if the sign changes and you multiply through then the direction of the inequality must change. No such problem with the equal part though.
1/(x - 1) + 3/(x - 4) = (-1/2)

1(x - 4) + 3(x - 1) = (-1/2)(x - 4)(x - 1)
x - 4 + 3x - 3 =(-1/2)(x^2 - 5x + 4)
4x - 7 =(-1/2)(x^2 - 5x + 4)
8x - 14 = -x^2 + 5x - 4
x^2 + 3x - 10 = 0
(x + 5)(x - 2) = 0
This gives you x = -5 and x = 2.
So these are the points where the equality part of the inequality holds true.

So let's put this all together. There are four points of interest (-5,1,2 and 4 and five regions of interest:
-INF to -5; -5 to 1, 1 to 2, 2 to 4 and 4 to INF.
let us examine each of these regions in turn:

(-INF, 5] is good since x=-5 give -1/2=-1/2 and values less than -5 give a value > -1/2
(-5,1) gives values < -1/2 so this is out. Look at 0, it gives-74>=-1/2 which is not true.
(1,2] is good. Look at 1.5 giving: 4/5 > -1/2.
(2,4) not good. look at 3: -5/2 >= -1/2 is not true.
(4,INF) this is also good since both terms on the left are greater than 0.

So the answer is:
(-INF,-5] U (1.2] U (4,INF)

Answer 2

The tricky part of inequalities is that you reverse them when you multiply both sides by a negative number:

Ex
2>1 but -2<-1

That's what's critical about -1 and 4 because under certain condictions you're reversing the inequality.

Presumably, you want to mulitiply by the product of the denominators:
(x+1)(x-4)
so that there are 4 regions to look at:

-inf to -1
-1 to 4
4 to inf

If you have a graphing calculator then graph
y1 = the left hand side
and
y2 = -1/2

and look at what happens.

Answer 3

1/(x+1) + 1/(x-4) >(or = )-1/2
multiply it by LCD
2(x+1)(x-4)
(x+6)(x-1)=0 Answer is -6 & 1
But (x+6)(x-1)>0 Answer is x>1

Answer 4

its should drow and consider about limit
for explian it u need graphs



ok thanks