How to solve a system of inequalities without graphing?

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Answer 1 · 2015-03-31T03:16:59

Here is one approach to solving it without graphing.

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

#{(3x-y <= 5),(x+y >= 1):}#

The system is equivalent to

#{(y >= 3x-5 ),(y <= -x+1):}#

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

#{(x,y)|y >= 3x-5}# (the set of all pairs #(x,y)# with #y >= 3x-5#)
and
#{(x,y)|y >= -x+1}# (the set of all pairs #(x,y)# with #y >= -x+1#)

Combining these we write:

#{(x,y)|(y >= 3x-5) " and " (y >= -x+1)}#.

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.