Linear Inequalities in Two Variables

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Chapter 7.1

• Linear Inequalities in Two Variables

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• Equation of a line yaxc
• if you graph any such line you can get
• horizontal lines
• slant lines
• but never vertical lines (example x2)
• we need more flexibility for the coefficient of
  y, so we allow for a b instead of just 1

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• Upgraded version of the equation of a line
  would look like this byaxc
• a slight modification (just a question of form)
• the above equation gives all possible lines
  horizontal ones (a0), vertical (b0), slant
  (neither zero)

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• Luckily, most of the problems well encounter
  will be for slant lines, in which case we can
  reduce this new form to the old one
• (b is not 0, so we can divide by it)

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• If the line still happens to be vertical (b0)
  this is usually a case easily solved
• Yet in definitions youll have to bear this
  upgraded form - such as the following

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• Definition A linear inequality in the variables
  x and y is an inequality that can be written in
  the form
• How to solve this inequality? We need to find ALL
  the points (x,y) which plugged in the equation
  produce the inequality

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• Example
• we start solving such a problem by first graphing
  the equation obtained by replacing the inequality
  with equality

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• Coefficient of y is not zero, so lets solve for
  y
• and now we can graph it
• VERY IMPORTANT POINT if the original inequality
  was STRICT, draw the line dotted if it was NOT
  STRICT, draw it normally (or even bold)

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• The reason is for strict inequalities the points
  on the line will not be solutions!
• On the other hand, for when we allow equality,
  the points on the line will be part of the
  solution

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• Graph of linear equation

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• If you now try to plug in points from one side or
  the other of the line, youll notice something
  points in the part of the plane (halfplane) where
  the origin lies, all are part of the solution
  points in the other halfplane are not.
• In fact, this is how you finish your proof if
  the origin is not on the line, check to see if it
  satisfies the inequality if it does, the

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• halfplane it lies in, is the solution if it
  doesnt, the other half is.
• Lets check for our problem the origin is not on
  the line, so plug (0,0) in
• hence the halfplane with the origin is the
  solution.

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• You can, in fact, formulate your answer in a
  similar manner to the following the halfplane
  generated by the line 2x3y60, containing the
  origin, together with the line (inequality is not
  strict) is the solution.
• what to do when the origin is on the line? Just
  choose another point which is not

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• And heres how you graph the solution

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• (for example, the point (30,20), which is in the
  other halfplane, doesnt satisfy the inequality,
  as you can check)

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• The cases when the line is vertical or horizontal
  are similar, with the exception that they are
  easier to draw!
• Examples/reminders y5 is the horizontal line
  intersecting the y-axis in 5 x10 is the
  vertical line intersecting the x-axis in 10

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• Graphs

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• One last point sometimes the original inequality
  is not simplified, with the unknowns scattered
  all over the equation for those you first have
  to simplify and bring the inequality to the
  standard form
• Example

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• We can now push the analysis further by
  considering systems of inequalities two, three
  or more inequalities at the same time
• as you can guess, we produce for each inequality
  its halfplane, then intersect all of them

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• Example

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• We have three lines
• and since the inequality is strict, draw it
  dotted
• inequality is not strict, so bold line
• strict inequality, so dotted again

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• The three lines

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• Take the first line check the origin, is it a
  solution? 0gt3 false, hence the solutions are in
  the halfplane that doesnt contain the origin

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• Take the second line the origin IS on the line,
  so take another point take (1,0), on the x-axis
  1gt0 true, so now we have the halfplane for the
  second inequality

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• We take the intersection of the two halfplanes

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• Go for the third line, and use again the origin
  0gt1/2 false, so its the halfplane that doesnt
  contain the origin

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• And again, take their intersection

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• Can it happen we have no solution? Sure! If the
  two lines are parallel, and the halfplanes are
  opposite, their intersection is empty

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• Graph
• intersection - none

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Chapter 7.2

• Linear Programming

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• Suppose you lead a company, and you want to
  maximize profit, or minimize cost. You have a
  number of constraints (scale of production, raw
  materials cost and usage, machinery availability,
  etc)
• If the formula you come up with for the profit
  (or cost) is linear, and the constraints are all
  linear inequalities, a popular method is called
  linear programming

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• The way it works is the following the
  constraints will build a region in the
  xy-plane, which has lines as boundaries
• the profit (or cost) function is a line, which is
  allowed to move, and, usually, the higher it
  is, the bigger the profit (the lower it is, the
  lower the cost)

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• The connection between the region and the line
  the line MUST intersect the region (profit/cost
  must satisfy the constraints)
• the point that allows for the highest position
  of the profit line is the solution - that is, the
  proper combination of raw materials, machinery
  and workers usage, etc, which maximizes profit
  (lowest for cost, minimizes cost)

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• An usual situation

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• Lets analyze the constraints first
• as said, they will only be linear inequalities,
  in two variables
• some constraints are explicit in the problem
• Example there are 180 hours available to work on
  a certain machine this machine can produce
  widgets type A, one per hour, or widgets type B,
  one per two hours the formula here is in terms
  of quantity of widgets A - x - and widgets B -y

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• Other constraints may not be explicitly given in
  the problem - they are natural conditions, and
  usually refer to single variables alone
• Example going back to our widgets, one cannot
  produce negative quantity of widgets, hence we
  have the following two extra constraints

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• The above example is by far the most common - it
  even has a name nonnegativity condition but
  more exotic problems might involve some other
  natural conditions (think of a problem where x is
  the number of fingers you must use - say, for
  typing then x must be less than 10 or 20 if
  you want to use your toes as well)
• The moment you have collected all the constraints
  its time to start solving them - and this is
  where you use what you learned in the previous
  chapter

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• One thing to notice about constraints is that the
  inequalities are all not strict - which means we
  dont worry whether we draw dotted or regular
  lines they are all regular (it means, the
  boundaries are part of the solution)
• there are three types of solutions (as per what
  we discussed in chapter 7.1) - by the way, the
  solutions are also called feasible regions

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• Bounded solution (most problems are like that)
  interpretation finite resources, or finite
  outputs - both variables, x and y, have a biggest
  value they cannot surpass
• Example take the above example, with the widgets

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• The solution

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• Empty solution interpretation no matter what
  combination of resources you choose, the
  requirements (the constraints!) wont be met - in
  real life this asks for re-thinking the strategy
  solving the actual mathematical problem looks
  like the last example given in chapter 7.1
• Example trivial one - a certain branch of a
  national, widgets producing company, has to
  produce at least 1000 widgets per day but state
  regulations wont allow for more than 500 widgets
  being produced per day call x the number of
  widgets - the example is trivial since you can
  see the answer right away

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• But heres the solution

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• Lastly, unbounded solutions interpretation
  resources/outputs are infinite - in practice one
  encounters cases when, for the moment, resources
  are in plentiful supply, or outputs can be
  produced in mass quantities these cases can be
  approximated by infinity mathematically, there
  are no bounds for either x or y, or for both of
  them
• Example A paper manufacturing company is
  producing white paper, and part of the quantity
  is passed through a dying process to make it
  colored.

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• Denoting the original quantity of white paper by
  x, and the quantity of colored paper by y, find
  all solutions of a feasible situation (we only
  have natural conditions x and y positive, and
  you cannot have more colored paper than you had
  originally white)

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• Draw the corresponding lines, check the possible
  halfplanes, point out the intersection
• as you can see, the solution goes to infinity

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• Lets now talk about the profit function
  (similarly for cost)
• usually, formulas for profit can be constructed,
  but they will rely on several factors, and will
  have quite complex expressions
• on the other hand, either by approximations, or
  by design, some profits can be expressed in terms
  of two factors, in a linear way

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• In what follows, as said at the beginning, we
  only handle profits which look like this
• notice that there is no value this expression is
  equal to - profit can be any number, in fact
  this is the reason its called profit function

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• On the other hand, we can choose ourselves a
  value for the profit (for example, your company
  wants to get a certain profit - equal the profit
  function with that value) and we can draw the
  result (we are looking for what pairs give us
  that profit)
• Example for each widget type A produced, a
  company has a net profit of 3 for each widget
  type B, 5 what is the profit function? For what
  quantities of widgets A and B do we get

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• a profit of 500?
• Profit3x5y
• draw the function 3x5y500 or, solving for y,
  y-0.6x100
• all points (pairs) on the line give us 500 profit

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• The above line is called isoprofit curve (line) -
  comes from the greek work isossame, hence
  same profit
• a nice feature for linear profits is that, when
  drawing the isoprofit lines for several values of
  the profit, we get parallel lines (try the
  previous examples expression for profit of
  1500, 3000, etc)

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• What is important to deduce is which direction
  does it increase (the profit)

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• You can find this direction by drawing at least
  two isoprofit lines, and noticing which line
  corresponds to the biggest profit
• same discussion for cost

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• Now, lets put the constraints AND the profit
  together
• the constraints will give you a region, only
  place where you can have a solution isoprofit
  lines can intersect the region or can avoid it -
  the second case is not good (interpretation
  impossible to obtain profit)

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• The ultimate goal is to find the biggest profit
  since you know which way the isoprofit lines go
  when increasing the profit, move an
  constraint-region-intersecting line THAT way, and
  keep going until you reach the boundary thats
  where you have to stop, so thats where you find
  your biggest profit

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• General practical strategy draw the constraint
  region draw two isoprofit lines, so you can see
  which way the profit increases draw a line
  parallel to those two isoprofit lines which
  passes through the region (unless one of the two
  already does intersect it) move it mentally in
  the direction of increasing profit, until you
  have to stop (if it so happens) draw that line

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• Remember, we have three types of regions we can
  optimize the above general strategy
• bounded notice that a maximum profit ALWAYS
  happens at a corner? (it must happen at the
  boundary, and the boundary is either a corner or
  a segment but a segment contains corners too)
  What about just checking the corners in the first
  place? There will be no need to draw any
  isoprofit lines no need to see which way the
  profit increases no need to move any lines

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• there will be, though, need to find the
  corners
• Example the widgets example - lets collect the
  constraints and the profit function

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• The graph
• how do we check the corners? Plug in the profit
  the values of x and y in each corner, and see
  which one gives you the biggest value

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• We have three corners
• (0,0) profit000
• (0, 180) profit0900900
• (90, 0) profit2700270
• biggest profit 900 where (0,180) - need to
  produce only widgets of type B, namely 180 of them

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• Empty regions - nothing here! No profit possible
  at all, since the situation is impossible to
  start with
• unbounded two things can happen here, namely
  either one of the corners is a solution OR the
  profit can increase indefinitely best way to
  handle this situation is to check the corners
  nevertheless, but also draw two isoprofit lines,
  to check the direction of increase of profit
  then decide whether the corners ARE maximums, or
  not

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• Example take the paper companys situation, with
  the extra thing profit is 1 for white paper and
  2 for colored (remember x was the original
  quantity of white paper, while y is part of that
  quantity which becomes colored, hence remaining
  white is y-x)

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• The graph
• only one corner, (0,0) with profit000

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• Draw two isoprofit lines, xy100 (y-x100) and
  xy200 (y-x200) on top of our graph
• since we can increase profit indefinitely, there
  is no maximum profit