The Graph of an Inequality

  • From: Christine McCormack
  • Date: 12 October 1999
  • Subject: Graphs of Inequalities

I have this question and I am unsure what it means and how to go about answering it:

Sketch the regions in the positive quadrant of the (x,y)-plane defined by the inequality constraints:

  1. 2x < 3y
  2. 3-6x > y
  3. 2x < 3y   AND   3-6x > y

Maths Help suggests:

What the question means

The first inequality,  2x < 3y   is equivalent to asking:
  "What numbers x and y will mean that
   2 times x is less than 3 times y ?"

Working out to confirm that 2x < 3y for x=2.5, y=2

There are infinitely many solutions to this question, for example  x = 2.5, y = 2

Because all of these solutions come as pairs of numbers for x and y, we can write them as coordinate pairs.

The question is asking you to colour in a region on a graph
where all the pairs of x, y coordinates satisfy the inequality.

How to answer the question

The region described by the inequality 2x<3y

The boundary of the region  2x < 3y  is a straight line.
The equation of this straight line is  2x = 3y.

To plot this straight line, make y the subject:
3y = 2x
  y = 2/3x

The region  2x < 3y  is everything to the left of the line  2x = 3y  
(To the left because 2x is less than 3y.)

Note that in your case you only need to sketch the graph in the positive quadrant,
which means only for positive values of x and y.

The answer to the second part is the region above the line  y = 3 - 6x:

The answer to the third part is the region where both of the inequalities are satisfied:

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