Application of Linear Inequality in Real Life – SS2 Mathematics Lesson Note (Third Term)

A linear inequality in two variables, x and y, compares the expression ax + by with a constant c. Any pair of numbers (x, y) that satisfies it is a solution, and all the solutions form a region of the x-y plane. Linear programming applies these graphs to real-life problems with restrictions such as limited money, storage or labour.

Behavioural objectives

By the end of the lesson, students should be able to:

• State the forms of a linear inequality in two variables.
• Check whether a pair of numbers is a solution of an inequality.
• Rewrite an inequality with y in terms of x and find the values of x and y where its boundary line meets the axes.
• Explain what linear programming is and name some restrictions it deals with.
• Write the inequalities for a real-life problem and use their solution set to answer questions about it.

What is a linear inequality in two variables?

A linear inequality in two variables x and y has one of these forms: ax + by ≤ c, ax + by < c, ax + by > c or 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 less than 17. But the pair x = 2 and y = 3 is not the only solution. There are infinitely many solutions.

If the pair of numbers x and y is a solution, think of this pair as a point in the plane. The set of all solutions can then be thought of as a region in the x-y plane. To determine this region, first express y in terms of x in the inequality.

5x + 2y < 17

2y < -5x + 17

y < (-5x + 17)/2

When x = 0, y = 8.5. When y = 0, x = 3.4. (Show this in a graph.)

What is linear programming?

In many real-life situations in business and commerce, there are restrictions or constraints that 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.

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 ₦500. She buys pencils at ₦50 each and erasers at ₦20 each. She gets at least five of each, and the money spent on pencils is over ₦100 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 ₦50 and y erasers at ₦20.

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:

50x – 20y > 100

5x – 2y > 10 (4)

The solution set of the four inequalities is given by the twelve points marked inside the shaded region. For example, the point (7, 6) shows 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 the point (6, 9).

Evaluation

1. Write the four forms of a linear inequality in two variables.
2. Show that x = 2 and y = 3 is a solution of 5x + 2y < 17.
3. Express 5x + 2y < 17 with y in terms of x, and find y when x = 0.
4. What is linear programming? Give two examples of restrictions that it deals with.
5. In Example 1, show that the point (7, 6) satisfies all four inequalities.

Assignment

1. Explain in your own words why the solutions of 5x + 2y < 17 form a region of the plane and not a single point.
2. In Example 1, explain how the inequality 5x – 2y > 10 comes from the sentence “the money spent on pencils is over ₦100 more than that spent on erasers”.