Find all the values of x in the interval [0, 2(pi)] that satisfy the inequality: 8cos(x)+4>0. The answer should be in interval notation.

Expert Answers

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8*cos(x) + 4 > 0

First we will subtract 4 from both sides.

==> 8*cos(x) > -4

Now we will divide by 8.

`==> cos(x) > -4/8`

`==> cos(x) > -1/2`

`==> cos(x) + 1/2 > 0`

But we know that `cos(x) = -1/2 iff x= (2pi)/3, (4pi)/3`

 ==> [ 0, 2pi/3 ] ==> cos(x) > 0

==> [ 2pi/3 , 4pi/3] ==> cos(x) < 0

==> [ 4pi/3 , 2pi] ==> cos(x) > 0

Then, we notice that the solution to the inequality is the interval [2pi/3 , 4pi/3]

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