Solving ax + b is less than cx + d

Date: 12/11/2001 at 23:54:35
From: Sara
Subject: Solving ax + b < cx + d

Here is a problem:

   6-9x is less than or equal to 5x-1

Please teach me the process in English, as if you were explaining to a 
10-year-old.

Date: 12/12/2001 at 08:50:36
From: Doctor Rick
Subject: Re: Solving ax + b < cx + d

Hi, Sara,

I assume you are learning algebra, but I'll write as if you had never 
heard of algebra.

Imagine that you have a balance scale - one of those things with two 
pans hanging by chains from opposite sides of a rod. "Justice" is 
often pictured holding one of these. You put things on each pan, and 
if the right side goes down, then what's in the right pan is heavier 
than what's in the left pan.

Now, we put 6-9x in the left pan and 5x-1 in the right pan, and the 
right pan goes down - or maybe the pans balance, I can't see it that 
clearly. In any case, the left pan DOESN'T go down! This is what "less 
than or equal" means: the left side is NOT heavier than the right 
side.

  6-9x <= 5x-1

Now, if I add the same amount to each pan, it won't change which pan 
is heavier. Let's add 1 to both sides. We get

  6-9x+1 <= 5x-1+1

The left side is the same as 6+1-9x (we can move the 3 things in the 
pan around without changing the total weight). Combine the 6 and 
the 1: it's 7-9x.

On the right side, -1+1=0, so all we have is 5x. Thus our balance now 
has

  7-9x <= 5x

Let's do the same thing again, only this time I will add 9x to each 
side.

  7-9x+9x <= 5x+9x

On the left, even though we don't know how much 9x is (since we don't 
know what number x stands for), we do know that -9x+9x = 0, just as 
-1+1 = 0. On the right side, we have 5 somethings and 9 somethings; 
regardless of what "something" is, that's 14 somethings. (This is the 
distributive property, for a 13-year-old!) Now we have

  7 <= 14x

The last step is harder to visualize with a balance scale. We have to 
take what is in each pan, and break it into 14 equal pieces. (Each 
of the pieces in the right pan will be x.) Then if all 14 pieces in 
the left pan weigh no more than all 14 pieces in the right pan, we 
know that one of the pieces from the left pan will also weigh no more 
than one of the pieces from the right pan. In other words, we can 
divide each side by 14:

  7/14 <= 14x/14

  1/2 <= x

Now I'm talking to a 13-year-old again. The one tricky part in working 
with inequalities is that, unlike when solving equations, you aren't 
allowed to divide or multiply by a negative number. Actually, you can 
do this, but you have to switch the direction of the inequality. If 
this confuses you, you can always find a way to solve the inequality 
without multiplying or dividing by a negative number. I did this by 
moving the -9x over to the other side, so that I didn't have a 
negative coefficient of x.

Does this help? Feel free to ask me about anything that still doesn't 
make sense to you.

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