The book tells about the relation of systems of linear
inequalities to convex polyhedra, gives a description of the
set of all solutions of a system of linear inequalities,
analyses the questions of compatibility and
incompatibility; finally, it gives an insight into linear
programming as one of the topics in the theory of
systems of linear inequalities. The last section but one
gives a proof of duality theorem of lienar programming.
The book is intended for senior pupils and all amateur
mathematicians.

The book was translated the Russian by Vladimir Shokurov and was
first published by Mir in 1979.

PDF | Cover | Bookmarks | OCR | 7.3MB | 130 pp | 600 dpi

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Contents

1. Some Facts from Analytic Geometry 8
2. Visualization of Systems of Linear Inequalities In Two or Three Unknowns 17
3. The Convex Hull of a System of Points 22
4. A Convex Polyhedral Cone 25
5. The Feasible Region of a System of Linear Inequalities in Two Unknowns 31
6. The Feasible Region of a System in Three Unknowns 44
7. Systems of Linear Inequalities in Any Number of Unknowns 52
8. The Solution of a System of Linear Inequalities by successive Reduction of the Number of Unknowns 57
9. Incompatible Systems 64
10. A Homogeneous System of Linear Inequalities. The Fundamental Set of Solutions 69
11. The Solution of a Nonhomogeneous System of Inequalities 81
12. A Linear Programming Problem 84
13. The Simplex Method 91
14. The Duality Theorem in Linear Programming 101
15. Transportation Problem 107

This booklet is one of longest in the LML series, having more than 120 pages. The back cover of the book says the following about the book.

The book tells about the relation of systems of linear inequalies to convex polyhedra, gives a description of the set of all solutions of a system of linear inequalities, analyses the questions of compatibility and incompatibility; finally, it gives an insight into linear programming as one of the topics in the theory of systems of linear inequalities. The last section but one gives a proof of the duality theorem of linear programming. The book is intended for senior pupils and all amateur mathematicians.

And the preface adds

Until recently one might think that linear inequalities would forever
remain an object of purely mathematical work. The situation
has changed radically since the mid 40s of this century when there
arose a new area of applied mathematics -linear programmingwith
important applications in the economy and engineering. Linear
programming is in the end nothing but a part (though a very important one) of the theory of systems of linear inequalities.
It is exactly the aim of this small book to acquaint the reader
with the various aspects of the theory of systems of linear inequalities, viz. with the geometrical aspect of the matter and some of the methods for solving systems connected with that aspect, with certain purely algebraic properties of the systems, and with questions of linear programming. Reading the book will not require any knowledge beyond the school course in mathematics.

The book was translated from the Russian by Vladamir Shokurov and was first published by Mir in 1979. You can get the book here.

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The following is the table of contents

CONTENTS
Preface 7
1. Some Facts from Analytic Geometry 8
2. Visualization of Systems of Linear Inequalities In Two or Three Unknowns 17
3. The Convex Hull of a System of Points 22
4. A Convex Polyhedral Cone 25
5. The Feasible Region of a System of Linear Inequalities in Two Unknowns 31
6. The Feasible Region of a System in Three Unknowns 44
7. Systems of Linear Inequalities in Any Number of Unknowns 52
8. The Solution of a System of Linear Inequalities ‘by Successive Reduction of the Number of Unknowns 57
9. Incompatible Systems 64
10. A Homogeneous System of Linear Inequalities. The Fundamental Set of Solutions 69
11. The Solution of a Nonhomogeneous System of Inequalities 81
12. A Linear Programming Problem 84
13. The Simplex Method 91
14. The Duality Theorem in Linear Programming 101
1.5. Transportation Problem 107

Day 1:  I split the kids up into groups of 2 and gave them each a group of a set of matching cards. I personally put the graphs on Green paper and the answers of Yellow paper (I am a Baylor Bear, so Green and Gold all the way!) I had a total of 18 graphs, and 18 answers. Some of the answers where in y=mx+b form, and others I wrote as slope and y-intercept, and had the students write the formula on the answer card. Finally, the students matched the graphs with their answers. Once the students had finally matched all the graphs, I asked them how to identify the slope, y-intercept, what are the steps of graphing a linear equation? This helped the students organize their thoughts, and helped me to get an understanding of what the students understand.

Matching Game Worksheets: Linear Equations Matching 1, Linear Equations Matching 2, Linear Equations Matching 3, Linear Equations Answers 1, Linear Equations Answers 2, Linear Equations Answers 3.

Day 2: The next day, I reviewed linear inequalities with the students. We talked about what types of lines (dashed or solid), and what direction the shading should be. After discussing the similarities and differences between linear equations and linear inequalities I gave the students the same graphs from the day before, but this time I changed the answers into linear inequalities. This gave the students the chance put to practice their understanding of inequalities. The students were required to match the graphs to the equation, and then add the changes that inequalities had on the graph (ie: where the shading could go, and if there should be a dashed or solid line). Using the same graphs as the day before gave the students an aspect of familiarity, and helped the students make connections between the linear equations and linear inequalities. I printed the inequality answers on blue paper… I personally see adding splashes of color adds a little extra something and creates a silly sense of excitement into the lesson.

Matching Inequalities Worksheets: *Use the same graphs as Day 1* Linear Inequalities Answers 1, Linear Inequalities Answers 2, Linear Inequalities Answers 3.

This was an excellent 2 day lesson that could be used to review or to teach the students about linear inequalities and linear equations.

Similarly, if I had asked “What can you tell me about absolute value?” most would have said something about changing negatives to positives. Few would have been able to give a satisfactory explanation if I had asked “What does |2 – 7| = 5 mean?”

After reading John Scammell’s recent post on linear inequalities, I realized that I had it backwards. I’d begin by graphing the line. I’d explain that a line cuts the plane into two regions. Together, we’d determine which region to shade. I’d tell students that each point in this region is a solution.

Instead, John’s colleague begins by having students find x- and y-coordinates that satisfy the inequality. Then, each student plots these ordered pairs on a grid at the front. It becomes clear to students that each solution is a point in a half-plane and that a boundary line exists. I robbed my students of this discovery.

John’s scatterplot reminded me of something from Kate Nowak’s back catalogue. Kate’s activity involves having students guess the number of M&Ms in a container. Plotting the points (guess, distance from correct value) results in something like this:

Kate has students write equations and inequalities that model weather forecasting. “Today’s temperature will be more than 10 degrees off from the usual temperature” can be modelled using |T – 68| > 10. In a later post, she has students write an inequality that models this scenario:

She scaffolds this by asking students to explain why |t – 12| ≤ 1993 and |t – 12| ≥ 1993 are not good models and by having students write a sentence that begins with “The distance from…”.

Context is important. Not because it must answer “When am I ever going to use this?” but because it helps build conceptual understanding. I’m guessing Kate’s students can tell her more about absolute value than “it changes negatives to positives”.

By the way, the inequality for music that I find tolerable would be |t – 1985| ≥ 6. I graduated from high school in ’91 not ’85. I’m just not a fan of 80s music.

(The title of this post was “borrowed” from Nat Banting.)

The common form of a Linear equation in the two variables like x and y is…..

y = mx + b where x and y are two variables and m and b are two constants.

The constant m determine s the slope or gradient of that line and the constant b shows the point at which line crosses the Y-axis. Constant b also known as Y-Intercept. Now we are going to discuss about Linear Inequality.In graphical manner, we can say that a linear inequality describes an area of the coordinate plane that has a boundary line. In simple way, in Linear Inequalities everything on one side of a line on a graph. In mathematics, a linear inequality is an inequality which involves a linear function. For solving inequalities we need to learn the symbols of inequalities like the symbol < means less than and the symbol > means greater than and the symbol ≦ or ≤ less than or equal to etc. Let’s see about some examples of Linear problems : -6 < x – 4 < 12

-6 < x – 4 < 12

add 4 to all 3 parts

-2 < x <16

divide 2 from all 3 parts

-2 < x < 16

To graph the following equation, you put an open circle or we can say mark it by a dot on the point (2,0) and then you put an open circle on the point (16,0).Then draw a line between the two.

Now we are going to discuss about linear equations. In simple mathematical manner, we can say that any equation that when graphed produces a straight line, then the equation is called as Linear Equation. The common form of a linear equation in the two variables like x and y is

y = mx + b

where x and y are two variables and m and b are two constants. The constant m determines the slope or gradient of that line and the constant b shows the point at which line crosses the Y-axis. Constant b also known as Y-Intercept.

Linear equations with inequality signs are called as linear inequalities. An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. A linear inequality describes an area of the coordinate plane that has a boundary line. In simple way in linear inequalities everything is on one side of a line on a graph. In mathematics, a linear inequality involves a linear function. For solving inequalities, we need to learn the symbols of inequalities like the symbol < means less than and the symbol > means greater than and the symbol ≦ or ≤ less than or equal to etc.

A system of equations come with the relationship between two or more functions, which can be used to form a number of real-world situations. It is basically a collection of two or more equations with a same set of unknowns. Systems of Equations Solvers are available online which generates answer on one click by user on solve option and also shows the student how to arrive at the answer.

‘via Blog this’

Introducing baby to algebra as early as the baby shower via algebra themed baby beginnings, such as:mobiles, room plaques, pacifiers and other baby algebra paraphernalia,we inundate baby with the message that algebra is important to baby and family tradition.Baby algebra uses pictures and key words to help Baby to generalize and grasp algebra concepts. Therefore we can think our way through the stepping stones called tests. Colors and images react . Colors with one side of the brain, images with the other side of the brain, together create and complete the learning process inherent at birth . WALLA! Baby does algebra.

Baby Algebra Is Here For Your Family

http://www.youtube.com/playlist?p=PLCE703DB5743508D7

• Video 1: Order of operations, algebraic expressions, linear equations, linear inequalities
• Video 2: Finding intercepts, graphing lines, finding slope, parallel lines, perpendicular lines, finding the equation of a line, linear inequalities in two variables
• Video 3: Systems of equations, substitution method, addition (elimination) method, coin problems
• Video 4: Exponent rules, negative exponents, add/subtract polynomials, multiply polynomials, square polynomials, polynomial division
• Video 5: Factoring polynomials, factor by grouping, trinomials, difference of squares, difference of cubes, solving quadratic equations by factoring, area of a rectangle
• Video 6: Simplify rational expressions, multiply/divide rational expressions, add/subtract rational expressions, solve rational equations, work-rate problems

If you’d like a copy of the 136 problem final review & answer key, drop me a line through the contact page on my website – georgewoodbury.com .

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

It can be a frustrating section for me as well. Some students really have a hard time grasping which half-plane to shade. The way I have always done it:

• Determine if the line is dashed or solid.
• Graph the line using an efficient technique.
• Pick a test point not on the line, preferably the origin (0,0).
• Substitute the coordinates into the original inequality for x and y.
  If the inequality is true, then the ordered pair is a solution. Shade the half-plane containing the test point.
  If the inequality is false, then the ordered pair is not a solution. Shade the half-plane that does not contain the test point.

Each semester there will be a group that can do everything except figure out which side to shade. It’s frustrating because I think it is pretty clear and straightforward.

Next semester I am changing my approach. Suppose the example is .

• I will have them graph the line, using the intercepts of (6,0) & (0,-4).
• We will then label the line and we will go over the idea that this line represents all of the ordered pairs for which is equal to 12.
• I will explain that this line divides the rectangular plane into two half-planes. One of the half-planes contains points for which and .
• I’ll pick a point below the line to the right, like (6,-5) and substitute the coordinates into the expression , which will result in 27. Since , we will then label that half-plane as .
• I’ll pick the origin, which is above the line to the left, and substitute the coordinates into the expression , which will result in 0. Since , we will then label that half-plane as .
• We will finish by shading the side labeled .

After 2 or 3 examples done this way, I think the students will understand what each half-plane represents and have an easier time determining which half-plane to shade.

How do you introduce this topic? How do you feel about this approach? Please leave a comment, or reach me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

The overall strategy is:
1) Graph the line – sometimes solid, sometimes dashed.
2) Pick a point not on the line, usually the origin (0,0), as a “test point”.
3) Plug the coordinates of the test point into the original inequality to determine if the “test point” is a solution or not.
4) Shade the side of the graph that contains solutions.

Now for a little more detail.
1) When you graph the line, the first thing you need to determine is whether the line will be solid or dashed.
If the inequality includes “equality” – <  or  > – then the line is solid because every point on the line is a solution.
If the inequality is a “strict” inequality, “<” or “>”, then we use a dashed line because the points on the line are not solutions of the inequality.

In terms of actually graphing the line:
If it is y = mx + b form, graph the line by plotting the y-intercept (b on the y-axis) and then using the slope (m) to find another point on the line.
If it is in Ax + By = C form, graph the line by finding the x-intercept (set y = 0) and the y-intercept (set x = 0). Put those two points on the graph and draw the line through them.
If it is of the form y = #, like y = 7, then this is a horizontal line.
If it is of the form x = #, like x = 2, then this is a vertical line.

2) Whenever the line does not go through the origin, pick (0,0) as your test point. We do this because plugging in 0 for x and 0 for y is as easy as it can be for us.
If the line goes through the origin, like y = 3x or 2x + 5y = 0, you have to pick another point besides the origin. I’d recommend a point on one of the axes so that at least you are plugging in 0 for one of the variables.

3) Plug the values for x and y into the original inequality.
If this produces a true statement, like 0 < 8 or 0 > -2, then the test point is a solution.
If this produces a false statement, like 0 < -6 or 0 > 5, then the test point is not a solution.

4) The line that you have graphed is like a street on a map, dividing the map into two neighborhoods. One side of the street contains all the solutions, and the other side is “abandoned” with no solutions.
If the test point led to a true inequality, then it is in the “good neighborhood” and we shade that entire neighborhood.
If the test point led to a false inequality, then it is in the “bad neighborhood”. We simply cross the street and shade the other neighborhood.

Do you have any questions? You can reach me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

The Good

My favorite part of teaching these inequalities is that it gives me a chance to go over efficient graphing techniques with students.

• If the equation is in slope-intercept form, graph using the y-intercept and the slope.
• If the equation is of the form , find the x- and y-intercepts and graph.
• If the equation only has one variable, then it is either horizontal or a vertical line .

Since the graphing exam is just around the corner, this is a jump-start on reviewing for the exam.

I also love these problems because of the critical thinking involved: determining the most efficient way to graph the line, determining whether the line should be solid or dashed, determining which half-plane to shade. This is the type of thinking required to be successful in an intro stat course and is a great preparation for students heading in that direction.

The Bad

Some students have a mental block when it comes to determining which side of the line to shade. I can walk them through the process of choosing a test point, and why the origin is a great choice if available. Most can substitute the coordinates of the test point into the original inequality, and even determine whether the inequality is true or false. But at that point it breaks down – “Which side should we shade?”

Don’t get me wrong. Many students understand this portion of the problem as well. But for the few who do not, it is a struggle trying to find a way to help those students to understand. I have tried

• Having students choose 2 test points, 1 on each side of the line. After substituting, I require them to write “False” or “True” next to each test point. Then we always “Shade the Truth!”
• Having the students write “True/False” next to their chosen test point. After substituting, they circle the correct word. If you circle “True”, shade that side of the line. If you circle “False”, move across the line and shade the opposite side.
• Having students that do understand explain their technique to classmates that are struggling. I do this in pairs or in groups up to size 4. (This is when the light bulb often turns on for students.)

Summary

In my experience I have to keep coming up with alternative explanations until I find one that each student understands. (In a separate blog I will lay out the general tips I offer my online students.)

Do you have any techniques that work for you and your students? Please share your experience and thoughts by leaving a comment, or reaching me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.