Inequalities – SS2 Mathematics Lesson Note (Second Term)

An inequality is a statement that compares two expressions using an inequality symbol such as < or >. Like an ordinary equation, an inequality has solutions. To solve one, add or subtract the same number or expression on both sides, or multiply or divide both sides by the same number, reversing the sign when that number is negative.

Behavioural objectives

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

• state the rules for solving inequalities;
• solve linear inequalities in one variable;
• represent linear inequalities on a number line;
• say what a solution of an inequality in two variables is;
• express an inequality in two variables with y in terms of x to find the boundary values of the solution region.

What are the rules for solving inequalities?

1. Add the same expression or number to both sides of the inequality, or subtract it from both sides, and keep the inequality sign.
2. Multiply or divide both sides of the inequality by the same positive number and keep the inequality sign.
3. Multiply or divide both sides of the inequality by the same negative number and reverse the inequality sign.

What is a linear inequality in one variable?

The expression 3x – 1 > x + 1 is a linear inequality in one variable x. A linear inequality in x is an inequality in which the highest power of x is one (unity).

How do you show a linear inequality on a number line?

A number line is used to illustrate linear inequalities in one variable. A point x = a divides the number line into 2 parts, x < a and x > a.

When x = a is included, the number line becomes:

A line segment from a to b is shown below.

How do you solve linear inequalities and show the answer on a number line?

Solve the following linear inequalities and represent each solution on a number line.

Example 1: 4x + 8 < 3x + 16

Subtract 8 from both sides:

4x + 8 – 8 < 3x – 8 + 16

4x < 3x + 8

Subtract 3x from both sides:

4x – 3x < 3x + 8 – 3x

x < 8

Example 2: 3(x – 6) < 9(x – 1)

Open the brackets:

3x – 18 < 9x – 9

Collect like terms:

3x – 9x < -9 + 18

-6x < 9

Divide through by -6 and change the sign:

x > 9/(-6)

x > -3/2

How do you solve inequalities in two variables?

A linear inequality in two variables x and y has one of these forms, where a, b and c are constants:

• ax + by ≤ c
• ax + by < c
• ax + by > c
• ax + by ≥ c

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

One solution to 5x + 2y < 17 is x = 2 and y = 3, because 5(2) + 2(3) = 16, which is less than 17. The pair x = 2 and y = 3 is not the only solution. There are infinitely many solutions. If a pair of numbers x and y is a solution, think of the pair as a point in the plane. The set of all solutions is then a region in the x-y plane.

To find this region, first express y in terms of x in the inequality:

5x + 2y < 17

2y < -5x + 17

y < -5x/2 + 17/2

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

Evaluation

1. What happens to the inequality sign when you multiply both sides by a negative number?
2. What makes an inequality a linear inequality in x?
3. Show that x = 2 and y = 3 is a solution of 5x + 2y < 17.
4. Express 5x + 2y < 17 with y as the subject.

Assignment

1. Solve 3x – 1 > x + 1 and show the solution on a number line.
2. Give two other pairs of numbers x and y that satisfy 5x + 2y < 17, and check each pair.