Question

solve the inequality express your solution using interval notation

x^4-7x^3+12x^2<0

Answer 1

y = x^2 * (x^2 - 7x + 12)
= x^2 (x-3)(x-4)

x^2 is always positive, so (x-3)(x-4) must be negative. That is true only for

3 < x < 4 or x in (3,4)

Answer 2

x^2(x^2 - 7x + 12) < 0
x^2(x-3)(x-4) < 0

critical values are x = 0, 3, 4
test for x< 0 , say x = -1
all we care about is the sign, so
(+)(-)(-) < 0 false
test for x between 0 and 3, says x = 1
(+)(-)(-) < 0 false
test for x between 3 and 4, say x = 3.5
(+)(+)(-) < 0 ---> TRUE
test for x > 4 , say x = 5
(+)(+)(+) < 0 false

so the solution is
3 < x < 4

the graph shown here, confirms this
http://www.wolframalpha.com/input/?i=x%5E4-7x%5E3%2B12x%5E2





http://www.wolframalpha.com/input/?i=x%5E4-7x%5E3%2B12x%5E2