How do you graph the inequality #–3x – 4y<=12#?

1 Answer
KillerBunny
Feb 5, 2015

To graph this kind of inequality, the best thing is to manipulate them and obtain a relation of the form #y\le f(x)# (or #y \ge f(x)#).

In fact, we know that #y=f(x)# is exactly the graph of #f#, and so #y\le f(x)# (or #y \ge f(x)#) represents all the portion of the plan below (or above) the graph of #f#.

Let's do those manipulations: starting from
#-3x-4y\le 12#,
adding #4y# at both sides we get
#-3x \le 4y+12#.
Subtracting 12 at both sides, we have
#-3x-12\le 4y#.
Dividing both sides by #4# we finally have
#-3/4 x - 3 \le y#
which can of course be read as
#y \ge -3/4 x - 3#

We thus have #f(x)=-3/4 x - 3#, which is a line and so it's very easy to graph. Once graphed, you need to consider all the portion of plan above the line to solve your inequality.

Here's the graph: graph{-3x-4y \le 12 [-10, 10, -5, 5]}

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