Graph of Linear Inequality in Two Variables – SS2 Mathematics Lesson Note (Third Term)

A linear inequality in one variable is an inequality in which the highest power of x is one, for example 3x – 1 > x + 1. Like an equation, it has solutions. You find them by adding, subtracting, multiplying or dividing both sides, and you reverse the inequality sign when you multiply or divide by a negative number.

Behavioural objectives

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

1. State what a linear inequality in one variable is.
2. Apply the rules for solving inequalities.
3. Solve linear inequalities in one variable.
4. Represent the solution of a linear inequality on a number line.

What is a linear inequality?

The term inequality applies to any statement involving one of the symbols <, >, ≤ or ≥. Similar to ordinary equations, inequalities have solutions.

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

What are the rules for finding the solutions to inequalities?

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

How do you show an 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 two 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.

Worked examples

Solve the following linear inequalities and represent them 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:

-6x/-6 > 9/-6

x > -3/2 (that is, x > -1½)

Evaluation

1. State the three rules for solving inequalities.
2. What is a linear inequality in one variable?
3. How does a point x = a divide the number line?
4. Solve 3x – 1 > x + 1 and represent the answer on a number line.

Assignment

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

1. 5x + 4 < 2x + 13
2. -2x < 8