Inequalities

Question

A) on the same set of coordinates axes, graph  the following system of inequalities

y<-2x+3

y-3x>-2

 

b)  write the coordinates of a point in the solution set of the inequalities graphed in part a

Answer 1

Graphing these is very much like graphing an equality.  Start by graphing them as an equality, that is y = -2x + 3.  That will tell you where the line actually goes first.  Graph the same way, i.e. using the intercepts, or just plugging in whatever you want for x and solving for y. 

 

Since it's not ≤ with the equal on it, the line you draw for that is going to be a dashed line _  _  _  _  _.  If there was the equal on it as well, it would be a solid normal line.  The dashed line shows where it is, but indicates it doesn't include the line.

 

Then you need to know which way of the line is the solution.  There are certainly other methods you can use to determine this, but one way is to "test" a point on either side of that dashed line.  Keep in mind, whichever side is correct, the whole side is the solution.  So if one point works in the inequality, then all points on that side work.  If the test point does not work in the inequality, then no points on that side will work, and you know it's the other side.

 

A really easy point to test is (0,0).  (Unless of course the line runs through that, then you pick something else.  And you don't have to use (0,0) - it's not some rule.  It's just easy.)

 

So 0 < -2(0) + 3

0 < 0 + 3

0 < 3

 

This is true, so (0,0) is a solution for that first inequality.  So whichever side (0,0) is on, that whole side is the solution.  So you might want to make a few light lines on that side to remind yourself which way it goes.

(Just for fun, also try the point (1,2).  You'll find it's a false statement, so that also indicates that whole side is not correct.  It should be, I hope it is, on the other side from (0,0). :-))

 

Then repeat everything with the other equation.

 

There will be a section between the dashed lines where both equations go - the only part that is a solution to both is the area when both of them go.  You shade that part in to show that's the solution of both.