Linear inequalities

CONTENT

-Application of Linear Inequalities in Real Life.

-Introduction to Linear Programming.

APPLICATION OF LINEAR INEQUALITIES IN REAL LIFE.

Greatest and Least Values

Example

Draw a diagram to show the region which satisfies the following inequalities.

5x + y ≥ – 4, x + y ≤ 4, y ≤ x + 2, y – 2x ≥ – 4

Find the greatest and the least value of the linear function F = x + 2y within the region.

Solution

For the inequality 5x + y ≥ – 4, first draw the line 5x + y = – 4.

When x = 0, y = 4, when x = -1, y = 1

Add a third point on your own and then draw line 5x + y = -4. You may need to extend the axes to do this:

Now use a test point such as x = 0, y = 0

When x = 0, y = 0, then 0 ≥ – 4 is true, so shade the region below the line 5x + y = -4.

For the inequality x + y ≤ 4, first draw the line x + y = 4.

When x = 0, y = 4 and when y = 0, x = 4.

So draw a line that passes through (0, 4) and (4, 0).

Test point: (0, 0), so 0 ≤ 4 is true. Shade the region above the line.

Similarly, for y ≤ x + 2 and y – 2x ≥ – 4, shade the unwanted regions.

The required region is labeled as R as shown. R is also called the feasible region (i.e. the region that satisfies a set of inequalities).

The greatest (maximum) and the least (minimum) of any linear function such as F = x + 2y occurs at the vertices (corner points) of the region which satisfies the given set of the inequalities.

At           A(-1, 1)                 F = x + 2y

                                                ⇒         F = -1 + 2 = 1

At           B(1, 3)                                   F = x + 2y

                                                ⇒            F = 1 + 6 = 7

At           C(2.67, 1.33)                       F = x + 2y

                                                ⇒            F = 2.67 + 2.66 = 5.33

At           D(0, -4)                 F = x + 2y

                                ⇒            F = 0 – 8 = -8

∴ F = x + 2y is least at the point D(0, 4).

    F = x + 2y is greatest at the point B(1, 3).

How to solve linear inequalities

Linear Programming

In many real-life situations in business and commerce there are restrictions or constraints, which can affect decision-making. Typical restrictions might be the amount of money available for a project, storage constraints, or the number of skilled people in a labour force. In this section we will see that problems involving restrictions can often be solved by using the graphs of linear inequalities. This method is called linear programming. Linear programming can be used to solve many realistic problems.

EVALUATION

1. A student needs at least three notebooks and three pencils. Notebooks cost N60 and pencils N36 and the student has N360 to spend. The student decides to spend as much as possible of his N360.
2. How many ways can he spend his money?
3. Does any of the ways give him change? If so, how much?
4. To staff a tailoring company, a businesswoman needs at least 6 cutters and 10 seamstresses. She does not want to employ more than 25 people altogether. To be effective, a cutter needs 2 tables to work on and a seamstress needs 1 table. There are only 40 tables available. If x and y are the numbers of cutters and seamstresses respectively,
5. Write down four inequalities that represent the restrictions on the businesswoman,
6. Draw a graph that shows a region representing possible values of x and y,
7. Find the greatest value of y

GENERAL EVALUATION/REVISION QUESTIONS

1. Draw the graphs of lines y=2x+1 and 2x+2y=7 on the same axes. Find the coordinates of their point of intersection to 1 decimal place.

2. Sketch the graph of the inequalities.

a.   3x+2>3      (b) 8-5x≤ 3        (c) 2x-3≤ 7

3. If x-6≤ 1 and 2x-1> 8, what is the range of values of x which satisfies both inequalities?

READING ASSIGNMENT

New General Mathematics SSS2, pages 98-111, exercise 10g.

WEEKEND ASSIGNMENT

Objectives

1. Given that x is an integer, what is the greatest value of x which satisfies 4-3x >24 ?

A. -7   B.  -6  C. -3  D.  6

2. Given that 3x+y =1 and x-7y=19, then x+y=   A. -2  B. -3    C. 5  D. 3

3. If 5+x≤7 and 4+x≥3, which of the following statement is true?

A. -3≤x≤3      B. -1≤x<3  C. -1≤x≤2    D. -1≥x≥2

3. Solve the inequality: 8-3x<x-4   A. x<-3  B. x<-4   C. x>3    D. x>4

4. The smallest integer that can satisfy the inequality 30-5x<2x+3 is    A. -4   B. 5   C. 3   D. 4

5.Solve the inequality 4y-7<2(3y-1)        A. y <  -5/2    B. y> -2/5     C. y< -5/3     D. y> -5/2

Theory

1. A supermarket gives a special offer to customers who purchase at least a pack of vests and a pack of T-shirts. The offer is restricted to a total of 7 of these items.

1. Write down three inequalities which must be satisfied.
2. Draw the graphs of the above conditions and shade the region that satisfies them.
3. If the supermarket makes a gain of N5 on each vest and N8 on each T-shirt, find the maximum gain made by the supermarket.

2. A man buys two types of printers. The table below shows the cost and the necessary working space required for each type.

Printer                  Cost                Working space

Type P                   N15, 000         4000 cm²

Type Q                  N25, 000         3000 cm²

The man has 48 000cm² of working space and he can spend up to N290, 000 to buy these machines.

1. Write down the inequalities to represent the above constraints.
2. Draw the graphs of these inequalities to show the feasible region.
3. Use your graph to find the maximum number of printers the man can buy.