Linear Inequalities - Mathematics Form 2 Notes

• Introduction
• Simple Statements
• Compound Statements
• Solution to Simple Inequalities
• Multiplication and Division by a Negative Number
• Simultaneous Inequalities
• Graphical Representation of Inequality
• Linear Inequality of Two Unknown
• Past KCSE Questions on the Topic.

Introduction

Inequality symbols

Statements connected by these symbols are called inequalities

Simple Statements

• Simple statements represents only one condition as follows
   
  • X = 3 represents specific point which is number 3
  • x >3 does not include 3 it represents all numbers to the right of 3 meaning all the numbers greater than 3 as illustrated above.
    
  • x < 3 represents all numbers to left of 3 meaning all the numbers less than 3.
    The empty circle means that 3 is not included in the list of numbers to greater or less than 3.
  • The expression x ≥ 3 or x ≤ 3 means that means that 3 is included in the list and the circle is shaded to show that 3 is included.

Compound Statements

• A compound statement is a two simple inequalities joined by “and” or “or.” Here are two examples.
  • 3 ≥ x and x > -3 Combined into one to form −3 < x ≤ 3
    
    All real numbers that are greater than − 3 but less or equal to 3
  • x > −6 and x < 3 forms −6 < x < 3
    
    All real numbers that greater than - 6 but less than 3

Solution to Simple Inequalities

Example

Solve the inequality
x − 1 > 2

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.

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
2x + 3 ≤ 5
Subtracting three from both sides gives
2x + 3 – 3 ≤ 5 − 3
2x ≤ 2
Divide both sides by 2 gives ^2x/₂ ≤ ²/₂
x ≤ 1

Example

Solve the inequality ¹/₃x − 2 ≥ 4

Solution

Adding 2 to both sides
¹/₃x − 2 + 2  ≥ 4 + 2
¹/₃x ≥ 6
¹/₃x × 3 ≥ 6 ×3
x ≥ 18

Multiplication and Division by a Negative Number

• Multiplying or dividing both sides of an inequality by positive number leaves the inequality sign unchanged
• Multiplying or dividing both sides of an inequality by negative number reverses the sense of the inequality sign.

Example

Solve the inequality 1 − 3x < 4

Solution

−3x – 1 < 4 – 1
−3x < 3
−3x > 3 
 −3    −3
Note that the sign is reversed x > −1

Simultaneous Inequalities

Example

Solve the following
3x − 1 > −4
2x +1 ≤ 7

Solution

Solving the first inequality
3x – 1 > −4
3x > −3
x > −1
Solving the second inequality
2x + 1 ≤ 7
2x ≤ 6 Therefore x ≤ 3 The combined inequality is −1 < x ≤ 3

Graphical Representation of Inequality

• Consider the following;
  x ≤ 3
   
• The line x = 3 satisfy the inequality ≤ 3, the points on the left of the line satisfy the inequality.
• We don’t need the points to the right hence we shade it
  Note:
• We shade the unwanted region
• The line is continues because it forms part of the region e.g it starts at 3.for ≤ or ≥ inequalities the line must be continuous
• For < or > the line is not continous, its dotted.This is because the value on the line does not satisfy the inequality.

Linear Inequality of Two Unknown

• Consider the inequality y ≤ 3x + 2 the boundary line is y = 3x + 2
  
  
• If we pick any point above the line eg (-3, 3 ) then substitute in the equation y – 3x≤ 2 we get 12 ≤ 2 which is not true so the values lies in the unwanted region hence we shade that region .

Intersecting Regions

• These are identities regions which satisfy more than one inequality simultaneously.

Example
Draw a region which satisfy the following inequalities y + x ≥ 1 and y − ¹/₂x ≥ 2

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 < x ≤ 2x + 5 on the number line
                           3
4. Solve the inequality x − 3 + x − 5 ≤ 4x + 6 − 1
                                     4          6           8
5. Solve and write down all the integral values satisfying the inequality.
   x – 9 ≤ − 4 < 3x – 4
6. 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 + 1 1
9. Given that x + y = 8 and x²+ y²=34
   Find the value of:-
   1. x²+2xy+y²
   2. 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 line 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. (3mks)
    
    
15. Write down the inequalities that define the unshaded region marked R in the figure below. (3mks)
    
    
16. Write down all the inequalities represented by the regions R. (3mks)
    
    
17. 1. 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 2cm to represent 1 unit on the x – axis and 1 cm to represent 1 unit on the y – axis. (6mks)
    2. State the turning point of the graph. (1 mk)
    3. Use your graph to solve.
       1. −x² + 3x + 4 = 0
       2. 4x = x2