Expert: Steve Holleran - 2/18/2008

Question
solve the  inequality and give in interval form: X^2+4X+4 divided by
X^2+4X > or equal to 0             Thank You!

Answer
Hi Kunjal,

The way I suggest you do this is to create a "sign chart":

First, factor the expressions:

       (x^2 + 4x + 4) / (x^2 + 4x)

=       (x+2)^2 / x(x + 4)

Now, on a number line, mark off the numbers that make each factor = 0"

---------|----------|-------|-----------

        -4         -2       0

Now pick numbers (test points) in each interval, and replace these numbers for x in the expression, and check the signs of each factor.
Now, If you notice, the numerator is a square, so you don't have to check them there--the result of squaring is always positive.  So, just check your numbers in the bottom factors: x and x + 4:

To the left of -4, let's test -5: x * x+4 = - * - = +, so put a + sign in this interval.

Between -4 an -2, let's test -3:  x * x+4 = - * + = -, so put a - sign here.

Between -2 an 0, let's test -1:  x * x+4 = - * + = -, so a - sign goes here,

Finally, to the right of 0, let's test 1:  x *x+4 = + * +, so a + sign goes here.

So your sign chart should look like:


      +             -           -           +
-------------|------------|-----------|-----------
            -4            -2          0

and now you can see your solution intervals are the ones with the + signs (greater than or = 0):

           (-inf, -4] U [0 , inf).

Hope you could understand what I did here.

Steve