Using Interval Notation to Express AnswersDate: 12/03/96 at 05:10:55 From: Juanita Johnston Subject: Algebra, complex fractions, and inequalities Solve for x and graph: x/(x+3) > or = 1/(x-1). The answer is also supposed to be given in interval notation. I've tried everything I can think of, and I just don't know where to start. I tried subracting the 1/(x-1) side, to get the x on the same side, and then combining the fractions I got it to look like: (x-3)(x+1) --------- > or = 0 (x+3)(x-1) I don't know where to go from there, or even if I'm on the right track! Date: 12/03/96 at 18:10:33 From: Doctor Mike Subject: Re: Algebra, complex fractions, and inequalities Juanita, Hey, you didn't leave very much for me to do! So far, it's perfect. Next you need to see that the numerator is zero for x=3 and x=-1, so the expression is equal to 0 for exactly those values. Also, the denominator is zero for x=-3 and x=1, so the expression is undefined for those values. (These values make the original version of the problem undefined, too.) If we stay away from those values, then none of the numerator or denominator factors can get to zero and cross over to the other sign. Let's draw a number line with these four special numbers marked: -x <------o----o----o----o----0----o----o----o----o------> +x -3 -1 1 3 The trick is to look at the sign (+ or -) of your expression in each one of the intervals where nothing changes sign from + to - or back. These intervals are "up to but not including -3", "strictly between -3 and -1", etc. There are 5 of them. As an example, let's see what happens between -1 and +1. The numerator has one positive factor and one negative factor. The same is true for the denominator. So both numerator and denominator are negative, making the fraction > 0. Now we know that it is >= 0 in the interval (-1,+1). Remember the fraction is 0 for x=-1 and undefined for +1, so we write [-1,+1); the square cornered bracket on the left shows -1 IS IN the interval we are talking about where the fraction is >= zero. With me so far? Now do this kind of analysis about the 4 other sub-intervals to see what happens. As for the graph for such a problem, just draw on the number line to show what x values are included. There are two ways to show that an end point is or isn't included in the graph. Some books use "[" and "(" for this and some use a filled-dot and hollow-dot for it. Look in your book. I hope this helps. Keep up the good work. -Doctor Mike, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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