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Testing inequalities on intervals

  1. Dec 21, 2011 #1
    How do you see if the following inequality holds true for (-2,0)?


    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?
  2. jcsd
  3. Dec 21, 2011 #2


    Staff: Mentor

    This is a quadratic inequality. The usual approach is to bring all the nonzero terms to one side so that the inequality looks like ax2 + bx + c < 0
    ax2 + bx + c > 0
    whichever is appropriate.
    The expression ax2 + bx + c can have no real roots, one real, repeated root, or two distinct real roots.

    If there are no real roots, the expression ax2 + 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 r1 and r2, the expression equals zero when x = r1 or when x = r2 and will change sign on either side of both roots. The two roots determine three intervals on the number line: (-∞, r1), (r1, r2), and (r2, ∞). 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.
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