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paulb203

Member

Registered: 2023-02-24

Posts: 261

Inequalities on Graphs

Does anybody else hate these?

Is it just me, or are they messy/awkward/UGLY/unsatisfying?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,524

Re: Inequalities on Graphs

Linear programming problems are very useful in the real world of business.  They're a way for a company to maximise its profits (or storage space or workforce etc).

I start by making a summary table of the constraints so I can construct the inequalities.  If an equation looks like this

then points on the line will satisfy ax + by = c

points one side of the line will satisfy ax + by < c

and on the other side      ax + by > c

So I start by drawing the line, then pick a random point on one side to see if it fits < or > .  If I can I choose a point with easy to work coordinates ... after all why pick a hard sum to do when an easy one will work just the same. 

Once all the constraint lines are drawn, they should enclose a polygon whose points are those satisfying all.  The thing you've then got to maximise or minimise will be another linear equation except this time we don't know 'c', that's the thing to maximise or minimise.   If you draw a random such line say ax + by = d, then any other line with the same form, ax + by = e, will be parallel.  So tracking across the polygon with parallel lines should enable you to pick out the 'best' point. It will always be a vertex.

You can also have non linear constraints. Don't know whether your syllabus includes those.  Same principles apply but the solution space isn't enclosed by straight lines.  It's then also possible that the 'best' point is on the curve rather than an end point.  I may have gone beyond what you need with this paragraph.

Why not post a past paper question?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 261

Re: Inequalities on Graphs

Thanks, Bob.

Here's the kind of thing we're doing;

https://www.mathsgenie.co.uk/resources/6-inequalities-regions.pdf

I'm on question 5 at the moment and finding it tricky

Bob

Administrator

Registered: 2010-06-20

Posts: 10,524

Re: Inequalities on Graphs

The constraints are all straight lines:

X + Y = 4
Y = 2X+1
Y=-1

If you draw those you'll find they define a triangle but it looks like the solution space is not the inside of it. You want below the first, above the second and above the third.

Later edit:

More on this.

These points lie on x+y =4
(0,4) (1,3) (3,1) (4,0)

These points are above the line
(0,5) (1,5) (3,5) (4,5)
For These x plus y is more than 4

Points like (1,-1) (3,-1) (4,-1) have x plus y less than 4 so  that's the side of the line that leads to the solution space.

(0,1) ((1,3) and (3,7) lie on y = 2x +1

We want y greater than 2x + 1 so choose points above this line like (0,5) ((1,5) etc.

But these points are outside the x +y <4 space.  Are there points that satisfy both?

Yes (0,3)) (-1,3) etc

Further edit.

I notice for question 6 that the inequalities also have an equals option. I think the convention is dotted line if points on the line are not allowed and solid line when they are.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob