In dealing with inequalities I've run into a certain peculiarity which I am currently unable to explain.
The example: Find the interval of time during which the ball is at least 32 feet above ground.
h = -16t^2 + 16t + 128 // Height of the ball in feet. -16t^2 + 16t + 128 >= 32 -16t^2 + 16t + 96 >= 0 -16(t^2 - t - 6) >= 0 -16(t+2)(t-3) >= 0 // At this point everything is going as planned.
// Now I have a choice to make (-16t - 32)(t - 3) >= 0 // This does not work OR (t + 2)(-16t + 48) >= 0 // This does work
// If I choose option 1, the relational operators are incorrect. -16t - 32 >= 0 and t - 3 >= 0 -16t >= 32 t >= 3 t <= -2
// If I choose option 2, the relational operators are correct. t + 2 >= 0 and -16t + 48 >= 0 t >= -2 -16t >= -48 t <= 3
Now, when looking at a graph it becomes obvious that the ball is at least 32 feet above ground during the interval [0, 3] (assuming time is not negative). Therefore, option 2 provides the correct relations for t, while option 1 inverts the relations.
What I don't understand is why this is happening, since multiplication is an associative operator. Seems to me that it shouldn't matter whether the -16 is multiplied into the first factor or the second, and yet it does. I would love to know why, so that I might circumvent this issue the next time around.