Date: Tue, 2 Mar 1999 17:48:25 -0600 (CST)
Sender: (Mallory White)
Subject: Pre-Algebra Question

Name: Mallory White

Who is asking: Student
Level: Middle

If the Problem was -4a plus -5 is less than or equal to 14, why would you change the sign to greater than or equal to?

Hi Mallory

If the change you talked about bothers you then you don't need to use it. Start with

-4a - 5 < 14.

You know that you can add or subtract the same number to both sides of an inequality and still have a true statement, so you can add 4a to both sides. This gives

-4a -5 + 4a < 14 + 4a

which becomes

-5  < 14 + 4a.

Now you can subtract 14 from both sides and you see

-5 - 14 < 14 + 4a  - 14

which becomes

-19 < 4a.

Now you can divide both sides by 4 and you get

-19/4  < 4a/4

which becomes

-4.75  < a.

Notice that if you read from left to right (like we do in many countries) you read  "-4.75 is less than a." If we read from right to left (like people do in many Asian and Arabic countries) you read  "a is greater than -4.75.". Mathematics is a universal language and we can read it either way.  So you can write the answer as -4.75 < a or a  > -4.75.
   What is illustrated here is that if you have an inequality with a negative number in front of the variable, you can deal with it by adding the quantity with a plus sign in front to both sides and effectively move the variable to the other side with a positive sign. However you may prefer to do the question the way your teacher probably showed you, many times it is faster. As an illustration, you know that 4 < 7 but -4 > -7. You can get from the first inequality to the second by the method above. Start with

4 < 7

and add -4 to both sides. This gives

4-4 < 7-4


0 < 3

Now add -7 to both sides to get

-7 < -4

or, reading from right to left

-4 > -7

It certainly quicker just to notice that if you multiply both sides of an equality by a negative number the direction of the inequality reverses.

Jack and Harley


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