# How do you find the solution to x - y > 3 and x + y < 3?

Apr 5, 2015

A solution to a system of two inequalities with two variables $x$ and $y$ is a set of pairs $\left(x , y\right)$ that satisfy both inequalities.

If you are looking for a solution to these two inequalities in terms of two separate inequalities, one for $x$ and another for $y$, you will not find them, there are no such solutions. Variables $x$ and $y$ are related and we cannot separate them.

The best approach to "solve" this system of two inequalities is to better represent all the pairs $\left(x , y\right)$ that satisfy them.

Let's represent the solutions to this system of inequalities (that is, all pairs $\left(x , y\right)$ that satisfy them) graphically.
First of all, let's transform both inequalities in a more graph-friendly representation.
Add the same number $y - 3$ to the first inequality obtaining
$x - y + y - 3 > 3 + y - 3$ or
$x - 3 > y$ or
$y < x - 3$
Add the same number $- x$ to the second inequality obtaining
$x + y - x < 3 - x$ or
$y < - x + 3$

Now we have two inequalities:
$y < x - 3$ and
$y < - x + 3$

Let's graph them now.
The graph of $y = x - 3$ is
graph{x-3 [-6, 6, -4, 2]}

All pairs $\left(x , y\right)$ that satisfy the equality $y = x - 3$ are on the line of this graph.
All pairs $\left(x , y\right)$ that satisfy the inequality $y > x - 3$ are above the line of this graph.
All pairs $\left(x , y\right)$ that satisfy the inequality $y < x - 3$ are below the line of this graph.
So, all the solutions to the first inequality are those pairs $\left(x , y\right)$ that lie below the line of this graph.

Similarly, the graph of $y = - x + 3$ is
graph{-x+3 [-6, 6, -4, 2]}

All pairs $\left(x , y\right)$ that satisfy the equality $y = - x + 3$ are on the line of this graph.
All pairs $\left(x , y\right)$ that satisfy the inequality $y > - x + 3$ are above the line of this graph.
All pairs $\left(x , y\right)$ that satisfy the inequality $y < - x + 3$ are below the line of this graph.
So, all the solutions to the second inequality are those pairs $\left(x , y\right)$ that lie below the line of this graph.

Notice that both graphs intersect at a point $\left(0 , 3\right)$.

Now we have to imagine an area on the coordinate plane that is both below the first graph AND below the second graph. This area represents a solution to a system of two inequalities we have.

You can consider this area as a quarter of a plain with a vertex at a point $\left(0 , 3\right)$ and borders diagonally going SE and SW from it.
Here is a graphical representation of this area with solutions to our system of inequalities being all the pairs $\left(x , y\right)$ representing all points below the following graph line.
graph{-|x-3| [-6, 6, -3, 3]}

Incidentally, you can express this with another inequality:
$y < | x - 3 |$