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

1 Answer
Apr 13, 2015

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.