Question 40620
Solve the rational inequality and state the solution set using interval notation.
{{{(x+3)/(x)<= -2}}}
Ok, first we need to eliminate the fraction {{{(x+3)/(x)}}} by multiplying both sides of the equation by the denominator x. 
This will create an equation that we can work with more easily.
{{{x((x+3)/(x))<= -2(x)}}}
Now we can simplify by canceling out the x's on the left side and multiplying on the right.
{{{x+3<= -2x}}}
Now we can subtract the x from both sides of the equation...
x + 3 - x = 3 and -2x -x = -3x
so...
{{{3<= -3x}}}
Now, this may get a little tricky...
We need to isolate the variable x. In order to do that we need to divide both sides by -3.
However, the rule for inequalities is that when we divide by a NEGATIVE number, we ALWAYS have to chanbge the inequality sign. so...
{{{3<= -3x}}} 3/-3 = -1 and 3x/3 = x 
And remember to change the sign.
So...
{{{-1>= x}}}
So this tells us that our variable x will always be less than or equal to -1.
The bracket ] tells us that -1 is included.
So we show this in interval notation...
(-~,-1] or (negative infinity, -1]
I hope this helps.
Good Luck!