Lesson BASICS - Graphing Inequalities

Source code of 'BASICS - Graphing Inequalities'

This Lesson (BASICS - Graphing Inequalities) was created by by longjonsilver(2297)

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About longjonsilver: I have a new job in September, teaching

Introduction
This discussion is focussed on straight line graphs. However, the theory can be applied to any type of graph.

I shall show you the very simple process of graphing inequalities and find the "region" bounded by several straight line inequalities.

Method
When asked to graph an inequality, first you plot (or sketch) the equality version, to get the straight line. Then we need to figure out which half of the graph is the required area.

So, lets look at an example...

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Example: Graph y < 3x + 1.
Solution
First we will plot the equation y = 3x + 1, to get the straight line. I am not going to teach you how to plot a graph, that is another Lesson. Hopefully, you can either plot it from coordinates, or if your maths skills are sufficient, you can sketch it by looking at the equation.

The graph is {{{graph(300,300,-5,5,-10,10,3x+1)}}}

Explanation: Now, if you walk along this line, every point is where y EQUALS 3x+1: that is what the equation said, and we have plotted that as a graph.

Now, how about the equation y < 3x+1? This is saying "where is y LESS THAN 3x+1. The answer will be either EVERYWHERE ABOVE THE LINE or EVERYWHERE BELOW THE LINE. The issue is how to figure out this.

THe way i started doing it was to pick a point either below or above the line... ANY point will do, but best if you pick an "easy" point. For that reason i pick one on the y-axis. Here, lets pick (0,0) the origin. The only thing to remember is pick a point that doesn't lie on the line.

So,
put (0,0) into y and 3x + 1
--> 0 and 3(0) + 1
--> 0 and 0 + 1
--> 0 and 1
so this says 0 is less than 1... just by looking at the numbers: zero IS less than one.

--> 0 < 1
so (0,0) is in the region y < 3x + 1, which is the region we were asked for...so we want the whole of the space BELOW the straight line.

To show this, shade the region above the line and write a key saying "unshaded region is the required region"

Question: why shade the region we don't want?
Answer: i shall show you with the next question.

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Example: Show the region bounded by {{{y<3x+1}}} and {{{y > -2x-1}}}.
Solution: We have already done the first equation.

Second equation - plot y = -2x - 1 on the same graph, and you get {{{graph(300,300,-5,5,-10,10,3x+1, -2x-1)}}}

So we already know that the region we want is below y=3x+1. We now need to figure out which part of that is satisfied by the second equation.

So again, pick a point not on the line, again i pick (0,0):

--> y and -2x-1
--> 0 and -2(0)-1
--> 0 and 0-1
--> 0 and -1
--> zero is greater than -1
--> y > -2x-1 which is the region we were asked for. This is the part above and to the right of y=-2x-1 (since this is where point (0,0) is, so shade the OTHER part, the part we don't want --> below and to the left... what remains is a region, unshaded, that is the region asked for in the question.

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Summary
Hopefully you can see from this, that the process is one that builds up. You have to do a certain amount of work, but that work is systematic and quite easy.

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Additional Example

The following example answers just the second part of the students' question.

*[problems 28678/15636, 28917/15843]