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For what values of \$z\$ is \$z^2-40z+340\le 4\$? Express your answer in interval notation.

Mar 17, 2020

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Problem:     z2 - 40z + 340  <=  4

z2 - 40z + 336  <=  0

(z - 12)(z - 28)  <=  0

There are three regions: the region from 12 to the left, the region from 12 through 28, and the region from 28 to the right.

Choosing a number in the region from 12 to the left:

-- I can choose any number: I'm going to choose 0 because it is easy to work with:

(z - 12)(z - 28)   --->   (0 - 12)(0 - 28)  =  (-12)(-28)  =  336

Since this answer is not <= 0, this region will not work.

Choosing a number in the region from 12 through 28; I'm going to choose 20:

(z - 12)(z - 28)   --->   (20 - 12)(20 - 28)  =  (8)(-8)  =  -64

Since this answer is <= 0, this region will work.

Choosing a number in the region from 28 to the right; I'm going to choose 30:

(z - 12)(z - 28)   --->   (30 - 12)(30 - 28)  =  (18)(2)  =  36

Since this answer is not <= 0, this region will not work.

So the answer is:  [12, 28]

Use square brackets because the problem has an 'equal-to' sign.

Mar 17, 2020