Multiplying or Dividing by a Negative Number
Date: 04/08/99 at 11:37:10 From: Ling-Seang Yap Subject: When x(x-4) < 0 When x(x-4) < 0 , if we divide (x-4) on both sides of this inequality, we get x < 0, but a negative x does not give a true answer for it. Example: x (x-4) < 0 x < 0 But if we substitute x with, say -4 -1 (-1 - 4) < 0 -1 ( -5 ) < 0 5 < 0 (false?) Hope you can shed light on this question. Thanks.
Date: 04/08/99 at 12:28:22 From: Doctor Peterson Subject: Re: when x(x-4) < 0 The problem in what you did is that when you multiply or divide an inequality by a negative number, it reverses the inequality. So if x - 4 is negative, when you divide by it it changes to x > 0. Since your example case of x = -4 makes x - 4 negative, you find that x(x-4) > 0. The best way to handle this kind of problem is to notice that there are two ways for the product x(x-4) to be negative: either x is negative and x-4 is positive, or x is positive and x-4 is negative. Let's take those cases one at a time: If x < 0 and x-4 > 0, then we have x > 4, which is impossible if x < 0. So this case gives no solutions. If x > 0 and x-4 < 0, then x < 4, and our answer is 0 < x < 4 That is, x(x-4) is negative when x is between 0 and 4. If we graph y = x(x-4), it is a parabola: * + * | + | * + * | + * | * + | * + * | *+ * | --+---+---*---+---+---+---*---+---+-- | +* * | + | * * + * * | * * + * | As you can see, it is negative just where we said. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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