Solve the linear inequality.

Remember how in basic algebra anything that we do to one side of an equation we do to the other as well to keep it balanced?    The same applies here with one caveat: if you multiply or divide by a negative number you must flip the inequality signs.  Here is an example of why this is important/

take 2<4, an obviously true statement and multiply both sides by -1 so we have -2<-4,  this is now not true!  If we flip the inequality so it says -2>-4 we have preserved out relation however. 

Secondly, its not so much about sides in algebra as it is about hitting every "chunk" of math that is separated by a relation symbol.  With that, back to your initial question.

Let's subtract four, not from both sides, but from all three "pieces"

-12<-2x<=-9      Now we divide by -2, dont forget to flip the signs!

6>x=>9/2      And we are done!

It can also be written as  x<6 and x>=9/2

I have a question. How do we get -2 when we subtract 4 from <4-2x ?

Am I correct in my assumption that you are talking about this line "-12<-2x<=-9 "

When we subtracted four it killed off the 4 in the 4-2x so what is left over is just the -2x in that middle "chunk"

Yes, that was the line I was speaking of. And now it all makes sense. Now I am going to try to do the next one.

Here's another inequality I want to turn into equalities.

Separately find the extreme values.

1. -8 = 4 - 2x₁

Add 8 to both sides:  0 = 12 - 2x₁

Add 2x₁ to both sides 2x₁ = 12

Divide by 2:  x₁ = 6 

2. 4 - 2x₂ = -5

Add 5 to both sides: 9 - 2x₂ = 0

Add 2x₂ to both sides: 9 = 2x₂

Divide by 2 to get x₂ = 4.5

The extreme values are 4.5 and 6. The only point to beware of here is that x can equal the second (lower) limit, but is strictly less than the first (upper) limit. So  4.5 ≤ x < 6 

It is good that you asked a further question sally. We want to know that you understand (or that you do not) 

Thumbs up for your clarification question.