If (x-4)(x+4)(x-7)≥0, we know that if x=4, -4, or 7 then the expression equals 0. After that, we want to know when x is greater than 0 by plugging in numbers between, after, and before these numbers. Before -4, we can plug in, for example, 5, to get (-9)(-1)(-12). Since there are 3 (an odd number) negative numbers, we know it is less than 0 and since -4 is the smallest number out of the 3 given (4, -4, -7), anything less than -4 is negative. Between -4 and 4, we could test 0 out, getting (-4)(4)(-7). Since there are 2 negative numbers (an even number of them), this is positive as is with anything between -4 and 4. Between 4 and 7, we can plug in 5, getting
(1)(9)(-2) which has one negative number (an odd number) which is therefore negative. For anything more than 7, we can plug 8 in, getting
(4)(12)(1) which has 0 negative numbers and is therefore positive.
Summing it up, anything less than -4 is negative, between -4 and 4 is positive, between 4 and 7 is negative, and anything higher than 7 is positive. Do you see the pattern? We started with something, and flip flopped till the end.
