Question

How would you solve the following inequality algebraically?



0.5(x + 2)^2 (x - 4) > 2x + 3

Answer 1

I would graph it to get close and look for intersections numerically.
You know that the cubic at left hits the x axis three times, grazes at x = -2 and crosses at x = 4
it starts at the lower left, y is big negative if x is big negative
then it bounces off the x axis at x = -2 and drips down to cross the y axis at y = -8 then goes up through x = 4 to x big + x and big +y on the right
the straight line on the right goes through (0,3) and (-3/2,0)
The curve crosses the line somewhere between x - -2and x = 0
You can see that the curve is above the line at the lower left but may cross for large negative x and it dips below at the lower right but might come up and cross agin for large + x
I would have to search for those crossings numerically.

Answer 2

ps
Once you have one root numerically, you can factor that out and be left with a quadratic.