A Company manufactures cell phone cases. The length of a certain case must be within .25 mm of 125 mm. All cases with Lengths outside of this range are removed from the inventory. How could you use an absolute value inequality to represent the lengths of all the cases that should be removed?

So, remember that:

cos(x) > 0   for  -pi/2 < x < pi/2

cos(x) < 0 for  pi/2 < x < (3/2)*pi

and

sin(x) > 0 for 0 < x < pi

sin(x) < 0 for -pi < x <0  or  pi < x < 2pi

Also, we have the periodicty of the sine and cocine equations, such that:

sin(x) = sin(x + 2pi)

cos(x) = cos(x + 2pi)

Now let's solve the problem:

here we have:

x = (13/36)π

This is larger than zero and smaller than π:

0 < (13/36)π < π

then:

Is positive.

The next one is:

Here we have x = (7/12)*pi

notice that:

7/12 > 1/2

Then:

(7/12)*π > (1/2)*π

Then:

is negative.

next one:

here:

x = (47/36)*π

here we have (47/36) > 1

then:

(47/36)*π > π

then:

is negative.

the next one is:

Here we have x = (17/10)*π

if we subtract 2*π (because of the periodicity) we get:

(17/10)*π - 2*π

(17/10)*π - (20/10)*π

(-3/10)*π

this is in the range where the cosine function is positive, thus:

is positive.

the next one is:

here we have:

x = (41/36)*π

Notice that both functions, sine and cosine are negatives for that value, then we have the quotient of two negative values, so:

is positive.

The final one is:

Here:

x = (5/9)*π

The sin function is positive with this x value, while the cosine function is negative, thus:

Is negative.