Name: Mallory White
Email: mallorywhite@hotmail.com

Who is asking: Student
Level: Middle

Question:
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

which becomes

Now you can subtract 14 from both sides and you see

which becomes

Now you can divide both sides by 4 and you get

which becomes

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

and add -4 to both sides. This gives

or

Now add -7 to both sides to get

or, reading from right to left

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.

Cheers,
Jack and Harley