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### Solving Inequalities: Cubic Curve, and Case by Case

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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.
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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
inequality.

Instead proceed as follows

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

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

x+5 - 6x + 2
--------------- < 0
(3x-1)(x+5)

(-5x+7)
------------  < 0
(3x-1)(x+5)

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
http://mathforum.org/dr.math/
```

```
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
positive.

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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Equations, Graphs, Translations

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