Associated Topics || Dr. Math Home || Search Dr. Math

### Why Flip the Inequality Sign?

```
Date: 10/26/2001 at 11:30:05
From: Sean
Subject: Flipping the Inequality sign

I have a question about an inequality problem:

Solve and graph 5 - 3x => 17. (=> is greater than or equal to).

Please tell me why you flip the inequality sign when dividing by a
negative number.

Thanks.
Sean
```

```
Date: 10/26/2001 at 12:09:56
From: Doctor Peterson
Subject: Re: Flipping the Inequality sign

Hi, Sean.

Let's try solving your inequality in two ways, one dividing by a
negative number and the other avoiding that.

First,

5 - 3x >= 17

-3x >= 12

x <= -4

Here I subtracted 5, then divided by -3 and reversed the inequality.

Next,

5 - 3x >= 17

5 >= 17 + 3x

-12 >= 3x

-4 >= x

This time I added 3x to both sides, eliminating the negative
coefficient; then I subtracted 17 and divided by 3. Notice that this
gives the same result, though it's written backwards. We know there's
nothing tricky in this method; so the reversal of the inequality must
be right when we do it the other way.

We can prove the general rule the same way:

if a >= b

then a - b >= 0  (subtracting b from both sides)

and -b >= -a     (subtracting a from both sides)

so that -a <= -b (rewriting the inequality in reverse)

This shows that when you multiply an inequality by -1 (which is part
of what you do when you divide by a negative number), you reverse the
inequality.

Here are some other discussions of this from our archives, which I
found by searching for the words "reverse inequality negative":

Negatives and Inequalities
http://mathforum.org/library/drmath/view/53142.html

Reversing the Inequality
http://mathforum.org/library/drmath/view/57463.html

Inequalities and Negative Numbers
http://mathforum.org/library/drmath/view/53177.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Negative Numbers
Middle School Algebra
Middle School Negative Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search