Question

How do you solve `2y - 2 + 6x < -4`?

Answer 1

Solution to an inequality like this is a set of all pairs `(x,y)` that satisfy it.
The right approach to this problem is to represent the solutions graphically.
First of all, let's simplify this inequality through a series of invariant (equivalent) transformations:
(a) add `2` to both parts of inequality:
`2y-2+6x+2< -4+2`
`2y+6x< -2`
(b) subtract `6x` from both parts of inequality:
`2y+6x-6x< -2-6x`
`2y< -2-6x`
(c) divide by `2` both parts of inequality:
`y < -2-3x`

The next step is to represent graphically the solutions to this inequality. To accomplish this, draw a graph of a corresponding equality:
`y = -2-3x`
graph{-2-3x [-10, 10, -5, 5]}
For every `x` a point on this graph with abscissa `x` the corresponding ordinate `y` equals to `-2-3x`. Those points that lie above this graph have the ordinate `y` greater than `-2-3x` and those points that lie below this graph have the ordinate `y` less than `-2-3x`, which is what we need.

Therefore, the area below this graph (not including the line itself) represents all the solutions to our inequality.
`y < -2-3x`
graph{y<-2-3x [-10, 10, -5, 5]}
The last inequality `y < -2-3x` is the algebraic solution, that might be expressed as "all pairs `(x,y)` that satisfy an inequality `y < -2-3x`", but the graphical representation seems to be better.