# Trig inequalities

#### EL ALEM

1. The problem statement, all variables and given/known data
find all values of x in the interval [0, 2pi] that satisfy the equation 2cosx + 1 less than or equal to 0

2. Relevant equations
None.

3. The attempt at a solution
for when 2cosx+1=0
cosx=-1/2
x= 2pi/3, 4pi/3

but what about the values for when 2cosx + 1 is less then 0? How to i fins those?

#### cyby

Since you have the roots - why don't you pick values between [0, 2pi/3), (2pi/3, 4pi/3), and (4pi/3, 2pi] to check? One or more of these intervals will give you your desired answer.

#### EL ALEM

Ok so i found 2cosx + 1 < 0 at this interval (2pi/3, 4pi/3) , so would that be my answer?

#### cyby

If that is the only interval, then indeed it is! In these types of problems, you can always divide the set into intervals and find test points...

#### EL ALEM

Thanks a bunch, one more question, if it was only less than instead of less than or equal to how would we divide the set into intervals?

#### cyby

Oh, well, technically speaking, if your inequality was $\leq$, then you should include the endpoints in the interval. If your inequality was $<$, then you should exclude the endpoints. So what you're really working with is [2pi/3, 4pi/3].

Ok thanks.

#### lasner12

Ok so i found 2cosx + 1 < 0 at this interval (2pi/3, 4pi/3) , so would that be my answer?

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