# How to solve a system of inequalities without graphing?

Mar 31, 2015

Here is one approach to solving it without graphing.

#### Explanation:

I think that you are asking about systems of inequalities in two variables, like

$\left\{\begin{matrix}3 x - y \le 5 \\ x + y \ge 1\end{matrix}\right.$

The system is equivalent to

$\left\{\begin{matrix}y \ge 3 x - 5 \\ y \le - x + 1\end{matrix}\right.$

We can write the solutions in "set-builder notation" as:

$\left\{\left(x , y\right) | y \ge 3 x - 5\right\}$ (the set of all pairs $\left(x , y\right)$ with $y \ge 3 x - 5$)
and
$\left\{\left(x , y\right) | y \ge - x + 1\right\}$ (the set of all pairs $\left(x , y\right)$ with $y \ge - x + 1$)

Combining these we write:

$\left\{\left(x , y\right) | \left(y \ge 3 x - 5\right) \text{ and } \left(y \ge - x + 1\right)\right\}$.

I think that for many people that way of writing the solution set is not as helpful as sketching the region of solutions in the coordinate plane.