find the values of x in the interval [0,2pi] that satisfy the inequality -7<7tan(x)<7

Answer 1

-7<7tanx<7

divide by 7

-1<tanx<1

-p/4<x<pi/4 and 3pi/4<x<5pi/4

or

-45<x<45 and 150<x<210 in degrees

but you want to convert the negative angles to positive angles less than 2pi

so,

0<x<pi/4 and 3pi/4<x<5pi/4 and 7pi/4<x<2pi

in interval notation

(0,pi/4)U(3pi/4,5pi/4)U(7pi/4,2pi)

Answer 2

-7 < 7tan(x) < 7

Divide both sides of each inequality by 7 to get:

-1 < tan(x) < 1

By being familiar with the unit circle, we know that tan(π/4), which is the same as tan(45°), equals 1 and also that tan(3π/4) equals -1, tan(5π/4) = 1 and tan(7π/4) = -1. So we can conclude that these π/4 multiples will be the dividing line that makes this true/false. There are also two point for which tan is undefined, which are π/2 and 3π/2 which should also be considered when dividing up the number line. With that, you can just make a number line and try out the various sections that these divide the number line into:

Now, try each section. π/4 is about 0.79, π/2 is about 1.57, 3π/4 is about 2.36, 5π/4 is about 3.93, 3π/2 is about 4.71, 7π/4 is about 5.5, and 2π is about 6.28

Section 1: Between 0 and π/4, I can try x = 0.5: tan 0.5 = 0.55 and this value is between 1 and -1 so 0 < x < π/4 is an interval that works.

Section 2: Between π/4 and π/2, I can try x = 1: tan(1) = 1.56 and this value is NOT between 1 and -1 so π/4 < x < π/2 is an interval that does NOT work.

Section 3: Between π/2 and 3π/4, I can try x = 2: tan(2) = -2.2 and this value is NOT between 1 and -1 so π/2 < x < 3π/4 is an interval that does NOT work.

Section 4: Between 3π/4 and 5π/4, I can try x = 3: tan(3) = -0.14 and this value is between 1 and -1 so 3π/4 < x < 5π/4 is an interval that works.

Section 5: Between 5π/4 and 3π/2, I can try x = 4: tan(4) = 1.16 and this value is NOT between 1 and -1 so 5π/4 < x < 3π/2 is an interval that does NOT work.

Section 6: Between 3π/2 and 7π/4, I can try x = 5: tan(5) = -3.38 and this value is NOT between 1 and -1 so 3π/2 < x < 7π/4 is an interval that does NOT work.

Section 7: Between 7π/4 and 2π, I can try x = 6: tan(6) = -0.29 and this value is between 1 and -1 so 7π/4 < x < 2π is an interval that works.

Sections 1, 4, and 7 all work. Putting these together, we get our solution as 0 < x < π/4 U 3π/4 < x < 5π/4 U 7π/4 < x < 2π