Question

solve the inequality in terms of intervals and illustrate the solution set on the real number line

x^3+2x<3x^2

Answer 1

start with
\[x^3-3x^2+2x<0\] then factor
\[x(x^2-3x+2)<0\]
\[x(x-1)(x-2)<0\]

Answer 2

the zeros of this are at 0, 1, and 2, so divide the real line in to four intervals
\[(-\infty, 0), (0,1), (1,2), (2,\infty)\]

Answer 3

it changes sign on each interval and if \(x>2\) all factors are positive, so the product is positive
therefore it is "negative" then "positive" then "negative" then "positive"
you want "positive" so pick the second and fourth intervals as your answer