Algebra question

Question

The profit formula for a small business is p = −x² + 160x − 6000.

 Find the set of x-values that will keep the profit positive. (Enter your answer using interval notation.)

Answer 1

First solve −x² + 160x − 6000 = 0

Factor it:

(-x + 60)(x - 100) = 0

Set each binomial equal to zero:

-x + 60 = 0 so x = 60

x - 100 = 0 so x = 100

Now, put these on a number line:

Notice that it breaks the number line into 3 sections. Now, try a number from each section to see if it makes the original condition true (we get a positive profit).

In the first section we can try x = 0:

p(0) = −(0)² + 160(0) − 6000 = -6000 (which is NOT a positive profit) therefore this section does NOT work.

In the second section we can try x = 80:

p(80) = −(80)² + 160(80) − 6000 = 400 (which IS a positive profit) therefore this section does work.

In the third section we can try x = 120:

p(120) = −(120)² + 160(120) − 6000 = -1200 (which is NOT a positive profit) therefore this section does NOT work.

So the answer is in between x = 60 and x = 100. To write this in interval notation: (60, 100)