Linear Inequalities - Mathematics Form 2 Notes

Share via Whatsapp


Introduction

Inequality symbols

 >   Greater Than
 ≥  Greater Than or Equal To
 <  Less Than
 ≤  Less Than or Equal To

Statements connected by these symbols are called inequalities



Simple Statements

  • Simple statements represents only one condition as follows
     simple inequalities
    • 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
      compound inequalities 1
      All real numbers that are greater than − 3 but less or equal to 3
    • x > −6 and x < 3 forms −6 < x < 3
      compound inequalities 2
      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/22/2
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
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
     graphical representation of inequalities
  • 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.

graphical representation of inequalities 2

Linear Inequality of Two Unknown

  • Consider the inequality y ≤ 3x + 2 the boundary line is y = 3x + 2
    two unknowns inequalities
  • 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

intersecting regions



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 − 54x + 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
    inequalities q10
  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
    inequalities q13
  14. Write down the inequalities that satisfy the given region simultaneously. (3mks)
    inequalities q14
  15. Write down the inequalities that define the unshaded region marked R in the figure below. (3mks)
    inequalities q15
  16. Write down all the inequalities represented by the regions R. (3mks)
    inequalities q16
    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
Join our whatsapp group for latest updates

Download Linear Inequalities - Mathematics Form 2 Notes.


Tap Here to Download for 50/-




Why download?

  • ✔ To read offline at any time.
  • ✔ To Print at your convenience
  • ✔ Share Easily with Friends / Students


Get on WhatsApp Download as PDF
.
Subscribe now

access all the content at an affordable rate
or
Buy any individual paper or notes as a pdf via MPESA
and get it sent to you via WhatsApp

 

What does our community say about us?

Join our community on:

  • easyelimu app
  • Telegram
  • facebook page
  • twitter page
  • Pinterest