x³ + 3x < 4x²
Subtract 4x² from both sides:
x³ - 4x²+ 3x < 0
Factor out an x:
x(x² - 4x + 3) < 0
Finish the factoring:
x(x - 1)(x - 3) < 0
Find the zeros of this inequality: x = 0, x = 1, and x = 3:
This divides the number line into four open-ended regions: (-∞,0), (0,1), (1,3), (3,∞)
(The regions are open-ended because the problem has a '<' sign; if there were a '≤' sign, square brackets would be used with the numbers, but not the infinity signs.)
Now test each region. Chose one number in the region: if it works, that region is part of your answer; if it doesn't work, that region is not part of your answer. You cannot choose an endpoint of the region!
For (-∞,0): I can choose any number in this region, I'll select -1 and test it:
---> (-1)(-1 - 1)(-1 - 3) = (-1)(-2)(-3) = -6. This is less than 0, so this region works.
For (0,1): I can choose any number in this region, I'll select .5 and test it:
---> (.5)(.5 - 1)(.5 - 3) = (.5)(-.5)(-2.5) = .625. This is greater than 0, so this region doesn't work.
For (1,3): I can choose any number in this region, I'll select 2 and test it:
---> (2)(2 - 1)(2 - 3) = (2)(1)(-1) = -2. This is less than 0, so this region works.
For (3, ∞): I can choose any number in this region, I'll select 4 and test it:
---> (4)(4 - 1)(4 - 3) = (4)(3)(1) = 12. This is greater than 0, so this region doesn't work.
Final answer: (-∞,0) ∪ (1,3)
For graphing purposes, place open circles at 0, 1, and 3 and shade the region to the left of zero and between 1 and 3.
