Inequalities

Date: 01/24/2001 at 22:08:56
From: Nick Kincannon
Subject: Pre-Algebra

How do you add, subtract, multiply, and divide inequalities?

Date: 01/26/2001 at 13:18:47
From: Doctor Alicia
Subject: Re: Pre-Algebra

Hi Nick,

When you add, subtract, multiply, and divide inequalities, you treat 
the inequality as if it were an equation (=). You flip the inequality 
if you divide or multiply by a negative number.

For example:

                       7 + 2x > 12

Subtract 7 from both sides:

                   7 - 7 + 2x > 12 - 7

                           2x > 5

Divide both sides by 2 but keep the inequality symbol as-is:

                           2x > 5
                          ---  ---
                           2    2

                            x > 5
                               ---
                                2

Therefore, x is greater than 5/2.  To check our work, pick any x 
greater than 5/2, such as 7/2.

                    7 + 2 * 7 > 12      
                           ---
                            2

The 2's cancel out:
                        7 + 7 > 12

                           14 > 12

So we see that our inequality holds true.

Now, let's try the a similar problem but one dealing with multiplying 
or dividing negative numbers:

                       7 - 2x > 12

Subtract 7 from both sides:

                   7 - 7 - 2x > 12 - 7

                         - 2x > 5

Divide by *negative* 2 and flip the inequality:

                         - 2x < 5
                          ---  ---
                         - 2   -2

The -2's cancel out:

                            x < -5
                                ---
                                 2

Therefore, x is less than -5/2.  To check our work, pick a number less 
than -5/2, such as -10.

                   7 - 2(-10) > 12

                       7 + 20 > 12

                           27 > 12

We see that the inequality holds true.

I hope this helps. Write back if you have any further questions.

- Doctor Alicia, The Math Forum
  http://mathforum.org/dr.math/