ucbugrad said:
How do you see if the following inequality holds true for (-2,0)?
(-x/4)*(x+2)>1
For that matter how do you test inequalities for a given interval in general?
Certainly there must be a way other than to check all values of (-x/4)*(x+2) in (-2,0) and see if they are greater than 1?
This is a quadratic inequality. The usual approach is to bring all the nonzero terms to one side so that the inequality looks like ax
2 + bx + c < 0
or
ax
2 + bx + c > 0
whichever is appropriate.
The expression ax
2 + bx + c can have no real roots, one real, repeated root, or two distinct real roots.
If there are no real roots, the expression ax
2 + bx + c is either always positive or always negative.
If there is one repeated root r, the expression equals zero when x = r and will be either always positive or always negative at all other values of x.
If there are two distinct roots r
1 and r
2, the expression equals zero when x = r
1 or when x = r
2 and will change sign on either side of both roots. The two roots determine three intervals on the number line: (-∞, r
1), (r
1, r
2), and (r
2, ∞). For problems in this category, it suffices to check any number from each of the three intervals. If the expression is negative at that x value, it will be negative for all other x values in that interval. Similarly, if the expression is positive at some point in one of these intervals, the expression will be positive at every other x value in that interval.