Mathematics Form 1-4 : CHAPTER THIRTY NINE – LINEAR INEQUALITIES

Specific Objectives

By the end of the topic, the learner should be able to:

1. Identify and use inequality symbols.
2. Illustrate inequalities on the number line.
3. Solve linear inequalities in one unknown.
4. Represent linear inequalities graphically.
5. Solve linear inequalities in two unknowns graphically.
6. Form simple linear inequalities from inequality graphs.

Contents

1. Inequalities on a number line.
2. Simple and compound inequality statements, e.g., x > a and x < b ⇒ a < x < b.
3. Linear inequality in one unknown.
4. Graphical representation of linear inequalities.
5. Graphical solutions of simultaneous linear inequalities.
6. Simple linear inequalities from inequality graphs.

Introduction

Inequality Symbols

Statements connected by these symbols are called inequalities. These symbols help us compare values and express relationships where quantities are not equal but have a specific order or range.

Simple Statements

Simple statements represent only one condition, as shown below:

X = 3 represents a specific point on the number line at 3. The inequality x > 3 represents all numbers to the right of 3, meaning all numbers greater than 3, as illustrated above. Similarly, x < 3 represents all numbers to the left of 3, meaning all numbers less than 3. The empty circle indicates that 3 is not included in the set of numbers greater or less than 3.

When the circle is filled or shaded, it means that the number (in this case, 3) is included in the set.

Compound Statement

A compound statement consists of two simple inequalities joined by “and” or “or.” Here are two examples:

Combined into one to form -3

Solution to Simple Inequalities

Example

Solve the inequality:

Solution

Adding 1 to both sides gives:

x – 1 + 1 > 2 + 1

Therefore, x > 3.

Note:

In any inequality, you may add or subtract the same number from both sides without changing the inequality’s direction.

Example

Solve the inequality:

x + 3 < 8

Solution

Subtracting three from both sides gives:

x + 3 – 3 < 8 – 3

x < 5

Example

Solve the inequality:

Subtracting three from both sides gives:

2x + 3 – 3 < …

Divide both sides by 2 to isolate x.

Example

Solve the inequality:

Solution

Adding 2 to both sides:

Multiplication and Division by a Negative Number

Multiplying or dividing both sides of an inequality by a positive number leaves the inequality sign unchanged.

However, multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. This is an important rule to remember when solving inequalities.

Example

Solve the inequality 1 – 3x < 4

Solution

-3x – 1 < 4 – 1

-3x < 3

Note that the sign is reversed: x > -1

Simultaneous Inequalities

Example

Solve the following:

3x – 1 > -4

2x + 1 < …

Solution

Solving the first inequality:

3x – 1 > -4

3x > -3

x > -1

Solving the second inequality:

Therefore, the combined inequality is:

Graphical Representation of Inequality

Consider the following:

The line x = 3 satisfies the inequality; the points on the left of the line satisfy the inequality.

We do not need the points to the right; hence, we shade that region.

Note:

We shade the unwanted region to indicate which values do not satisfy the inequality.

The line is continuous because it forms part of the region, e.g., it starts at 3. For inequalities, the boundary line must be continuous if the inequality includes equality (≤ or ≥).

If the line is not continuous, it is dotted. This indicates that the values on the line do not satisfy the inequality (strict inequalities < or >).

Linear Inequality of Two Unknowns

Consider the inequality y < 3x + 2; the boundary line is y = 3x + 2.

If we pick any point above the line, e.g., (-3, 3), and substitute into the inequality y < 3x + 2, we get 3 < 3(-3) + 2 = -7, which is false. Therefore, the values lie in the unwanted region; hence, we shade that region.

Intersecting Regions

These are regions that satisfy more than one inequality simultaneously. Draw a region which satisfies the following inequalities:

End of topic

Past KCSE Questions on the Topic

1. Find the range of x if 2 ≤ 3 – x < 5.

2. Find all the integral values of x which satisfy the inequalities:

2(2 – x) < 4x – 9 < x + 11.

3. Solve the inequality and show the solution:

3 – 2x ≤ 2x + 5 on the number line.

4. Solve the inequality x – 3 + x – 5 ≤ 4x + 6 – 1.

5. Solve and write down all the integral values satisfying the inequality:

x – 9 ≤ -4 < 3x – 4.

1. Show on a number line the range of all integral values of x which satisfy the following pair of inequalities:

3 – x ≤ 1 – ½ x

-½ (x – 5) ≤ 7 – x

7. Solve the inequalities 4x – 3 ≤ 6x – 1 < 3x + 8; hence represent your solution on a number line.

8. Find all the integral values of x which satisfy the inequalities:

2(2 – x) < 4x – 9 < x + 11.

9. Given that x + y = 8 and x² + y² = 34, find the value of:

a) x² + 2xy + y²

b) 2xy

10. Find the inequalities satisfied by the region labelled R.

11. The region R is defined by x ≥ 0, y ≥ -2, 2y + x ≤ 2. By drawing suitable straight lines on a sketch, show and label the region R.

12. Find all the integral values of x which satisfy the inequality:

3(1 + x) < 5x – 11 < x + 45.

13. The vertices of the unshaded region in the figure below are O(0, 0), B(8, 8), and A(8, 0).

Write down the inequalities which satisfy the unshaded region.

14. Write down the inequalities that satisfy the given region simultaneously. (3 marks)

15. Write down the inequalities that define the unshaded region marked R in the figure below. (3 marks)

16. Write down all the inequalities represented by the regions R. (3 marks)

17. a) On the grid provided, draw the graph of y = 4 + 3x – x² for the integral values of x in the interval -2 ≤ x ≤ 5. Use a scale of 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis. (6 marks)

b) State the turning point of the graph. (1 mark)

c) Use your graph to solve:

(i) -x² + 3x + 4 = 0

(ii) 4x = x²