Application Of Linear Inequality In Real Life SS2 Mathematics Lesson Note

SOLUTIONS OF INEQUALITIES OF TWO VARIABLES AND THE RANGE OF VALUES OF COMBINED INEQUALITIES

A linear inequality in two variables x and y is of the form: ax + by   c:  ax + by < c:    ax + by > c    ax + by  c where a, b and c are constants.  A solution to an inequality is any pair of numbers x and y that satisfies the inequality.

Example:

Determine the solution set of 5x + 2y  17.

Solution

One solution to 5x + 2y < 17 is x =2 and y = 3 because 5(2) + 2(3) = 16, which is indeed less than 17.  But the pair x = 2 and y = 3 is not the only solution.  There are infinitely many solutions.  If the pairs of numbers x and y are a solution, then think of this pair as a point in the plane, so the set of all solutions can be thought of as a REGION in the x –y plane. Hence, to illustrate how to determine this region, first express y in terms of x in the inequality.

3x + 2y  17

2y  -5x + 17

Y < -5x + 17

         2      2 

When x = 0, y = 8.5; when y = 0, x = 3  (show in a graph) 

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.

Example 1 

A student has N500. She buys pencils at N50 each and erasers at N20 each. She gets at least five of each and the money spent on pencils is over N100 more than that spent on erasers. 

Find     

1. How many ways the money can be spent, 
2. The greatest number of pencils that can be bought, 
3. The greatest number of erasers that can be bought. 

Solution 

Let the student buy x pencils at N50 and y erasers at N20.

From the first two sentences, 

50x + 20y < 500

5x + 2y < 50                 (1)

Since she gets at least five of each,

x 5                         (2)

y 5                        (3)

From the third sentence, 

5x – 2y > 10  

The solution set of the four inequalities is given by the twelve points marked inside the shaded region. For example, points (7, 6) show that the student can buy seven pencils and six erasers and still satisfy the restrictions on the two variables. Hence there are twelve ways of spending the money. 

The greatest number of pencils that can be bought is eight, corresponding to the point (8, 5)

The greatest number of erasers is nine, corresponding to points (6, 9).