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solve the inequalities.9
Example 10.7.3. Solve the following inequalities on [0, 2π ). Express your answers using interval
notation and verify your answers graphically.
1. 2 sin(x) ≤ 1 2. sin(2x) > cos(x) 3. tan(x) ≥ 3 Solution.
1. We begin solving 2 sin(x) ≤ 1 by collecting all of the terms on one side of the equation and
zero on the other to get 2 sin(x) − 1 ≤ 0. Next, we let f (x) = 2 sin(x) − 1 and note that our
original inequality is equivalent to solving f (x) ≤ 0. We now look to see where, if ever, f is
undeﬁned and where f (x) = 0. Since the domain of f is all real numbers, we can immediately
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set about ﬁnding the zeros of f . Solving f (x) = 0, we have 2 sin(x) − 1 = 0 or sin(x) = 2 .
π
5π
The solutions here are x = 6 + 2πk and x = 6 + 2πk for integers k . Since we are restricting
π
our attention to [0, 2π ), only x = π and x = 56 are of concern to us. Next, we choose test
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values in [0, 2π ) other than the zeros and determine if f is positive or negative there. For
x = 0 we...

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