1. TBF 122 - Generall Mathematics - II Lecture – 8 : Linear Inequalities
Prof. Dr. Halil İbrahim Karakaş
Başkent University

2. Linear Inequalities in two Variables.
Let us recal the definition of a linear equation in two variables:
If a, b and h are real numbers, then ax + by = h is called a linear equation in the variables x and y.
The numbers a and b are called coefficients, the number h is called the right hand side constant; the symbols x and y are called variables or unknowns.
Now a, b and h being real numbers, each of the expressions , , ,
İs called a linear inequality in the variables x and y.
The definitions of the terms coefficient, the right hand side constant and variable or unknown are valid for inequalities, too. .
Let x0 and y0 be real numbvers. Given a linear inequality, if the inequality obtained by substituting x0 for x and y0 for y is true, then the ordered pair (x0 , y0 ) is called a solution of that linear inequality. In that case we also say that the ordered pair (x0 , y0 ) satisfies the given inequality.
Consider the linear inequality 3x - 4y < 12. (2,7) is a solution of this inequality, because
32 - 47 < 12 is true.
The pair (7,2) is not a solution of that inequality, because the inequality 37- 42 < 12 is not true.

3. Like the graphs of linear equations in two varibles, one can consider graphs of linear inequalities .
Recall that the graph of a linear equation in two variables is a line. The graph of a linear inequality is completely determined by the line which is the graph of the corresponding linear equation. In fact, it is easy to observe that eveery line divides the plane into two half-planes. The graph of a linear inequality is one of the half-planes determined by the line corresponding to that linear inequality.
To clarify what we have said above, note that every vertical line divides the plane into two half-planes (see the figure). The half–plane which is left to that line is called the left half-plane and the half-plane right to the line is called the right half-plane. y
x= a
Left half-plane
Right half-plane x
If the inequality sign used is  or , the points on the line x=a are not included in the graphr; if the inequality sign is  or  , then the line x=a is included in the graph.
x  a
x  a

4. The half-plane which lies above an inclined line is called the upper half-plane and the half-plane which lies below is called the lower half-plane.
For each point (x , y) on the line ax + by = h, we have y = -(a/b)x + h/b (see the figure).
Upper half-plane y
For each point (x , y1) in the lower half-plane, we have y1 < -(a/b)x+h/b; for each (x , y2) in the upper half plane, we have y2 > -(a/b)x+h/b.
ax+by = h y2 y
If b > 0, then y1
y < -(a/b)x+h/b  ax+by < h ,
Lower half-plane x x
y > -(a/b)x+h/b  ax+by > h;
and thus the graph of ax+by < h is the lower half-plane; the graph of ax+by > h is the upper half-plane of the line ax+by = h.
If b < 0 , then
y < -(a/b)x+h/b  ax+by > h ,
y > -(a/b)x+h/b  ax+by < h
and thus the graph of ax+by < h is the upper half-plane; the graph of ax+by > h is the lower half-plane of the line ax+by = h. .

5. The graph of the inequality ax+by < h is
a) the lower half-plane of ax+by = h if b > 0 ,
b) the upper half-plane of ax+by = h if b < 0.
The graph of the inequality ax+by > h is
a) the upper half-plane of ax+by = h if b > 0,
b) the lower half-plane of ax+by = h if b < 0.
The graph of ax+by < h or that of ax+by > h does not include the line ax+by = h; the graph of ax+by  h or that of ax+by  h includes the line ax+by = h.
To determine which half-plane is the graph of a linear inequality, we may also use a test point which is not on the line corresponding to the given inequality. For instance, if it is not on the line corresponding to the given inequality, one may take (0,0) as the test point.
If the test point satisfies the given inequality, then the graph of the inequality is the half-plane which contains the test point; if the test point does not satisfy the inequality, then the graph of the inequality is the half-plane which does not contain the test point.

6. We summarize what we have obtained in a theoarem.
Theorem. Let a , b, h  ℝ.
1. If b > 0, then the graph of ax + by < h is the lower half-plane of ax + by = h; the graph of ax + by > h is the upper half-plane of ax + by = h.
2. If b < 0, then the graph of ax + by < h is the upper half-plane of ax + by = h; the graph of ax + by > h is the lower half-plane of ax + by = h.
3. If b = 0 and a  0, the graph one of ax < h and ax > h is the left half-plane of x = h/a and the graph of the other pone is the right half-plane of x = h/a .
Let a , b, h  ℝ. To sketch the graph of any of ax + by < h , ax + by  h , ax + by > h or ax + by  h one can follow the following steps:
1. Draw the line ax + by = h as a dashed line if the inequality sign is < or >, as a solid line if the inequality sign is  or  .
2. To determine which half-plane is the graph of the given equation
a) Use the above theorem or
b) Choose a test point( a point which is not on the line you have drawn, for instance (0,0) if it is not on that line). If the test point is a solution of the inequality, then the graph is the half-plane which contains the test point; if not, the graph is the opposite half-plane.
Now we give concrete examples.

7. Example. Consider the line 3x – 4y = 12.
Since b = – 4 < 0, the graph of 3x – 4y < 12 is the upper half-plane,
the graph of 3x – 4y > 12 is the lower half-plane. y
Test point
3x – 4y = 12
3x – 4y < 12
upper half-plane
(0,0)
(4,0) x
3x – 4y > 12
(0,-3)
lower half-plane
The same conclusions can be achived by taking (0,0) as a test point.
(0,0) is contained in the upper half-plane and it satisfies 3x – 4y < 12.

8. Example. The inequality 3x – 4y ≤ 12.
Since b = – 4 < 0, the graph of 3x – 4y ≤ 12 is the upper half-plane. y
Test point
3x – 4y = 12
3x – 4y ≤ 12
upper half-plane
(0,0)
(4,0) x
(0,-3)
The same conclusions can be achived by taking (0,0) as a test point.
(0,0) is contained in the upper half-plane and it satisfies 3x – 4y < 12.

9. x 3x – 2y = 6 (0,0) Example. The graph of 3x – 2y  6. y Test point
3.0 –2.0  6
b = – 2 < 0 x
upper half-plane
(0,0)
(2,0)
(0,-3)

10. Example. The graph of 2x – y > 6.
(0,0)
2x – y = 6
Test point
2.0–0 < 6
b = – 1 < 0
(3,0)
lower half-plane
(0,-6)

11. 0 > -2 (0,0) Example. Tha graph of y > - 2. y Test point x
(0,-2)
y = - 2

12. 2x = 3 Example. The graph of 2x  3. y Test point x (0,0) (3/2,0)
2·0 < 3 x
(0,0)
(3/2,0)
2x = 3

13. y x 0 < 3·1 x = 3y (0,1) (3,1) (0,0) Example. The graph of x  3y.
Test point
0 < 3·1
b = – 3 < 0
upper half-plane
x = 3y
(0,1)
(3,1)
(0,0) x

14. Convexity.
An obvious property of half-planes which can be proved very easily:
Given two points in a half-plane, the line segment joining these two points lies in that half-plane.
Let K be a set of points in the plane. İf the line segment joining any pair of points in K lies again in K , then K is said to be a convex set.
Every half-plane is a convex set.
It is not difficult to see that the intersection of a finite number of convex sets is again convex. Thus the intersection of a finite number of half-planes in a plane is convex.

15. Systems of Linear Inequalities.
A collection of two or more linear inequalities is called a system of linear inequalities.
An ordered pair of real numbers is called a solution of a system of linear inequalities if it is a solution of each linear inequality in the system.
Hence, solution set of a system of linear inequalities is the intersection of solution sets of the linear inequalities in the system. Solution set of a system of linear inequalities in two variables can be considered as a set of points in the plane; i. e, one can sketch the graph of a system of linear inequalities in two variables.
The graph of a system of linear inequalities is also called its solution region. Solution region of a system of linear inequalities in two variables is bounded by line segments or half-lines. These line segments or half-lines are called borders of the solution region. A line which contains a border of the solution region is called a border line of the solution region. A point of the solution region is called a vertex if it is the intersection of two border lines.

16. Example.
To sketch the graph of the system, we sketch each inequality and consider the intersection. y
For each inequality we first sketch the corresponding line and determine which half-plane the graph is and paint each half-plane with a different color.
(0,5)
x –4y=0
Instead of painting the half-planes, we may put arrows to indicate which half-plane is the graph of the inequality considered. Looking at these arrows it becomes easier to determine the solution region of the system.
(4,1)
(0,1) x
(0,0)
(5,0)
x+ y=5

17. Solution region has a corner point:y
x+ y=5
Solution region has a corner point:
the solution of
(0,5)
SOLUTION REGİON
Corner point
(4,1).
x –4y=0
(0,1)
(4,1) x
(0,0)
(5,0)

18. Corner points of the solution region:
Example. y
(5,6)
(0,20)
(9,2)
2x + 5y=40
(0,11)
(0,8) x
(0,0)
(11,0)
(10,0)
(20,0)
Corner points of the solution region:
x + y=11
(0,0) , (10,0) , (9,2) , (5,6) , (0,8)
2x+ y=20

19. Corner Pointss of the solution region:
Example. x y
(0,0)
(5,0)
(0,5)
(1,4)
4x – y=0
(4,1)
x – 4y=0
x+ y=5
Corner Pointss of the solution region:
(4,1) , (1,4)

20. Corner Points of the solution region:
Example. x y
(0,0)
3x+ y =15
(0,15)
(2,5)
x + 2y =12
(4,3)
(0,7)
(0,6)
(7,0)
(5,0)
(12,0)
Corner Points of the solution region:
x + y =7
(0,0) , (5,0) , (4,3) , (2,5) , (0,6)

21. The examples we had so far give two types of solution regions
The examples we had so far give two types of solution regions. Solution regions of some systems of linear inequalities can be enclosed within a circle while this impossible for solution regions of some other systems of linear inequalities. This observation leads to the following definitions.
A solution region of a system of linear inequalities is said to be bounded if it can be enclosed within a circle. A solution region which can not be enclosed within a circle is said to be unbounded.
(4,1) x y
(0,0)
x – 4y=0
(5,0)
(0,5)
x+ y =5
(1,4)
4x – y=0
(0,6)
(0,7)
x + y=7
(5,0)
3x+ y=15
x + 2y=12
(4,3)
(2,5) x y
(0,0)
bounded
unbounded

22. Consider a linear function defined by K(x,y) = Ax + By
Consider a linear function defined by K(x,y) = Ax + By. Does it have a maximum or minimum value on the solution regişon of a given system of linear inequalities? If so how to find it? We will study these questions in the rest of the present lecture.
The concept of convexity is important for answering these questions.
Recall that intersection of a finite number of half-planes is convex. Since the solution region of a system of linear inequalities is the intersection of a finite number of half-planes, it is a convex set.
Using the convexity of solution regions, one can prove the following theorem:
Theorem (Corner Point Theorem). If a linear function of two variables has a maximum or minimum value on the solution region of a system of linear inequalities, that maximum value or the mnimum value must occur at a corner point (or corner points) of the solution region.

23. Theorem (Corner Point Theorem)
Theorem (Corner Point Theorem). If a linear function of two variables has a maximum or minimum value on the solution region of a system of linear inequalities, that maximum value or the mnimum value must occur at a corner point (or corner points) of the solution region.
Let D denote the solution region of a system of linear inequalities and let (x0 ,y0)  D. Assume that K(x0,y0) = Ax0 + By0 = K0 is the maximum value of K(x,y) = Ax + By on D. Thus K(x,y) = Ax + By ≤ K0 for all (x,y)  D. In other words, D is contained in the half-plane given by the inequalityAx + By ≤ K0.
If (x0,y0) is the only point of D on the line Ax + By = K0 , then (x0 ,y0) is a corner point of D. Otherwise, a side of D is a part of the line Ax + By = K0 .
In both cases, the line Ax + By = K0 contains a corner point of D; the value of K(x,y) = Ax + By that corner point is its maximum value.
Clearly, a similar discussion can be given for the minimum value (if it exists) of the linear function K(x,y) = Ax + By on D.

24. Corner point theorem idea also about the existence of maximum or minimum value of a linear function on the solution region of a system of linear inequalities. For instance, if D is bounded, one can find numbers K1 and K2 such that D lies in the upper half-plane of Ax + By = K1 and also in the lower half-plane of Ax + By = K2 (see the figure below on the left). Thus if D is bounded, K(x,y) = Ax + By has both a maximum and a minimum value on D. y
(0,0) x
Ax + By = K2
Ax + By = K1 D x y
(0,0)
Ax + By = K1
If D is an unbounded region contained in the first quadrant and if A and B are both positive, (see the figure above on the right), then for a suitable number K1, the region D lies in the upper half-plane of Ax + By = K1, but D is never contained in the lower half-plane of any inclined line. Threfore, in that case, K(x,y) = Ax + By has a minimum value but it has no maximum value on D.

25. Thus we have proved the following theorem.
Teorem. Let D be the solution region of a system of linear inequalities and let K be a linear function of two variables defined by K(x,y) = Ax + By.
a) if D is bounded, then K(x,y) = Ax + By has both a maximum and a minimum value on D. .
b) If D is an unbounded region contained in the first quadrant and if A and B are both positive, then K(x,y) = Ax + By has a minimum value but no maximum value on D.
In part b) of the theorem, we have the condition that the coefficients A and B are both positive. This condition is equivalent to the condition that the slope of the line Ax + By = C is negative. In part a) of the theorem, there is no restriction on A and B.

26. Let us consider the linear function
Example.
Let us consider the linear function
K(x,y) = 4x + 5y on the solution region of x y
(10,0)
(9,2)
(5,6)
(0,8)
(0,0) D
corner
K(x,y)
(0,0)
(0,8) 40
(5,6) 50
(9,2) 46
(10,0)
As we see from the table, K(x,y) = 4x + 5y attains its maximum value on at the corner point (5,6) and it attains its minimum value at (0,0). Maximum value is K (5,6) = 50, minimum value is K(0,0)= 0.

27. Example.
Let us consider the linear functions K(x,y) = 4x + 5y and L(x,y) = 4x - 5y on the solution region of x y
(4,1)
(1,4)
(0,0)
Since the solution region D is an unbounded region contained in the first quadrant and the coefficients in K(x,y) are both positive, K(x,y) = 4x + 5y has a minimum value but no maximum value on D. The minimum value is K(4,1)=21.
As for the linear function L(x,y) = 4x-5y, one of the coefficients is negative; therefore we can not apply the theorem.
However we know that if L possesses maximum or minimum value on D, it should occur at the corner points. The values L(1,4)=-16 and L(4,1)=11 at the corner points can not be maximum or minimum values for L; because for any t ≥ 17, the points (t,t) and (3t,t) belong to the solution region and we have L(t,t)=4t–5t=-t < -16, L(3t,t)=12t–5t=7t >11.

28. Sometimes one might need to ask for maximum or minimum values on a solution region of a function of the form N(x,y) = Ax + By + C where A, B and C are real numbers. Denote the solution region by D and define K(x,y) = Ax + By .
It is easily seen that
If K(x,y) = Ax + By does not have a maximum (or minimum) value on D, then N(x,y) = Ax + By + C has no maximum (or minimum) value on D, too.
If K0 is the maximum value of K(x,y) = Ax + By on D, then K0 + C is the maximum value of N(x,y) = Ax + By + C on D.
If K1 is the minimum value of K(x,y) = Ax + By on D, then K1 + C is the minimum value of N(x,y) = Ax + By + C on D.

29. x y
(10,0)
(9,2)
(5,6)
(0,8)
(0,0) D
Example.
We can see by evaluating at the corner points or by adding 20 to the corresponding values of K(x,y) = 4x + 5y that the maximum and minimum values of N(x,y) = 4x + 5y + 20 on the solution region given on the right hand side are N(5,6) = 70 and N(0,0) = 20 , respectively. x y
(4,1)
(1,4)
(0,0)
N(x,y) = 4x + 5y + 20 has minimum N(4,1) = 41, and no maximum on the region given on the right hand side.
Example.

30. is called a linear equation in n variables.
Linear Inequalities in Several Variables. Let a1 , a2, ... , an  ℝ, n  2 and x1 , x2, ... , xn variables. Recall that the expression
is called a linear equation in n variables.
The expression obtained by replacing the equality sign, = , with any one of < , > , ≤ or  is called a linear inequality in n variables.
Thus each of the following is a linear inequality in n variables. , ,
n The terms coefficient, right hand side constant, variable used for linear equations are used for linear inequalities as well.
Let c1 , c2 , , cn  ℝ . If the numerical inequality obtained by substituting c1 for x1, c2 for x2 , , cn for xn in any of the above linear inequalities is true, then the ordered n-tuple (c1 , c2 , , cn ) is said to be a solution of that linear inequality.
If the number of variables is n=2, instead of x1 , x2 we may use the letters x , y; if n=3 , we may use the letters x , y , z instead of x1 , x2 , x3 .

31. Example.
Consider the linear inequality x1 + 2x2 - 3x3 + 4x4 < 15. The quadruple (1,2,3,4) is a solution of that inequality, because =12 < 15. On the other hand , (2,3,4,5) is not a solution of that inequality, because =16 > 15.
It is possible to visualize the solution set of a linear inequality in two variables as a set of points in the plane and similarly, it is possible to visualize the solution set of a linear inequality in three variables as a set of points in the space. However, it is not possible to do so if the number of variables is more than three.
As in two varibles case, one can think of systems of linear inequalities in n varibles for n 3. The solution region of such a system of linear inequalities is the intersection of solution sets of the linear inequalities in the system.
We can skech solution regions of systems of linear inequalities in two variables in the plane, but this becomes difficult or impossible when the number of variables is more than two. However, systems of linear inequalities in several variables are often used in constructing mathematical model of many real life problems. We shall have examples of these in the forthcoming lectures.