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Solving Inequalities: Cubic Curve, and Case by Case

Date: 07/13/99 at 17:08:59
From: tommy mandich
Subject: Solving Inequalities

1/(3x-1) < 2/(x+5)

Solve the inequality and express the solution in terms of intervals.

I know that you have to get the x's all on one side. When I do all the 
arithmetic I get x > 7/5. I know that x can't = 1/3 and -5.

The answer at the end of the book is (-5,1/3) U (7/5,infinite).

I'm not sure what to do.  

Date: 07/13/99 at 17:33:15
From: Doctor Anthony
Subject: Re: Solving Inequalities

You must not cross-multiply an inequality like this, because the 
expression by which you are multiplying might be negative for some 
values of x and that would require you to reverse the direction of the 

Instead proceed as follows

     1/(3x-1) - 2/(x+5) < 0

        x+5 - 2(3x-1)
       ---------------  < 0

        x+5 - 6x + 2
       --------------- < 0

        ------------  < 0

The easiest way to decide the range of values is to multiply the three 
brackets together, since whether you multiply or divide the rule of 
signs will be the same. So consider the cubic expression

   f(x) = (-5x+7)(3x-1)(x+5)

We require the values of x for which f(x) < 0, that is, the curve is 
below the x axis.

This cubic would have a -15x^3 term meaning that the cubic comes in 
from the left at y = + infinity cuts the x axis at x = -5 goes below 
for a short time, then turns up to cut the x axis again at 1/3 goes up 
for a short space, then turns down again to cut the x axis for the 
last time at x = 7/5 before disappearing off to -infinity as x goes 
to +infinity.

From this description we see that the curve is below the x axis from 
-5 to 1/3 and then again from x = 7/5 to infinity. So these ranges 
of x will satisfy the inequality.

- Doctor Anthony, The Math Forum   

Date: 07/13/99 at 17:34:56
From: Doctor Jaffee
Subject: Re: Solving Inequalities

Hi Tommy,

Here is how I approach problems like this:

There are four cases to consider. First, 3x - 1 and x + 5 might both 
be positive. If that is the case, you can multiply both sides of the 
inequality by 3x - 1 (which is positive) and x + 5 (which is positive) 
and this should lead to part of the solution.

Then consider what happens when 3x - 1 is negative and x + 5 is 

Next, consider 3x - 1 being positive and x + 5 being negative.

Finally, see what happens when 3x - 1 and x + 5 are both negative.

In each case, multiply both sides of the inequality by (3x - 1)(x + 5) 
and don't forget that when you multiply both sides of an inequality by 
a negative the order of the inequality changes direction.

Give this a try and see if you come up with the correct answer. If you 
get stuck, write back and I'll try to help you out some more.

Good luck.

- Doctor Jaffee, The Math Forum   
Associated Topics:
High School Equations, Graphs, Translations

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