Does anyone understand exactly what is going on in the example and how to use it for the next two problems? can anyone explain this value substitution method? !

From the looks of it, it is just smart substitution.  Basically, you want to pick values for "x" in such a way that one of the variables disappear.  Do this enough times and you can get a solvable set of equations.

So for (2)
Pick x = 1 (doing this removes the a variable)
We get:
4b + c = 1

Now pick x = -1 (doing this removes the b variable)
We get: 
4a + c = 1

Now there is no way to remove the c variable so let's just pick x = 0. 
We get:
a + b + c = 0

Now we have 3 set of equations and 3 variables.

Solve this by using Gauss-Jordan elimination or whatever other technique you know (Or just use an online matrix calculator).

You should get:
a = 1/2
b = 1/2
c = -1

first anwser choice.

step-by-step explanation:

the greater/less or equal to symbol (> or < with the line at the bottom) is a closed/filled in circle.

when multiplying or dividing by a negative on both sides you have to flip the inequality sign. this rule only applies to multiplying or dividing by a negative number in inequality equations. this rule is really important.

-5c + 2 _< _27

subtract from both sides by 2

-5c _< _ 27

divide both sides by -5

c _< _ 5.4

flip the inequality sign

c _> _5.4

the only possible answer that would match the simplified inequality is the first answer choice