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

Apr 13, 2015

Solution to an inequality like this is a set of all pairs $\left(x , y\right)$ 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:
$2 y - 2 + 6 x + 2 < - 4 + 2$
$2 y + 6 x < - 2$
(b) subtract $6 x$ from both parts of inequality:
$2 y + 6 x - 6 x < - 2 - 6 x$
$2 y < - 2 - 6 x$
(c) divide by $2$ both parts of inequality:
$y < - 2 - 3 x$

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

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