SUBJECT: MATHEMATICS

CLASS: SS 2

DATE:

TERM: 2nd TERM

REFERENCE BOOKS

New General Mathematics SSS2 by M.F. Macraeetal.
Essential Mathematics SSS2 by A.J.S. Oluwasanmi.

WEEK ONE

TOPIC: LINEAR INEQUALITIES IN ONE VARIABLE

CONTENT

-Linear Inequalities

-Inequalities with Reversing Symbols

-Representing the Solutions of Inequalities on a Number Line and on Graphs

-Combining Inequalities

LINEAR INEQUALITIES

There are different signs used in inequalities.

> Greater than

< Less than

≥ Greater or equal to

≤ Less or equal to

= Not equal to

Example 1

Consider a bus with x people in it.

(a)If there are 40 people then x= 40, this is an equation not inequality.â®

(b)If there are less than 30 people in the bus then x 30 where means less than ; this is an inequality. It literally means that the no of people in the bus is not up to 30.

Example 2

Find the range of value of x for which

7x – 6 ≥ 15

7x ≥ 15 + 6

7 x ≥ 21

x≥ 3

Example 3:Solve the inequality

12x -7≥ 13 + 2x

12x-2x≥13+7

10x≥20

x≥2

Evaluation

Solve the inequalities

3x -10 < 2

2.Given that x is an integer, find the three greatest values of x which satisfies the inequality 7x+15≥2x

Inequalities with Reversing Symbols

Anytime an inequality is divided or multiplied by a negative value, the symbol is reversed to satisfy the inequality.

Example

Solve: 14- 2a < 4

- 2a < 4 – 14

- 2a < -10

Divide both sides by -2 and reverse the sign (symbols).

a > 5

Check:

If a > 5,then possible values of a are : 6,7,8,…

Substituting, a=6

14 - 2(6) < 4

14 -12 < 4

2 < 4

2 2 - 3x ≤ 2(1-x)

Multiply through by 3 or put the like terms together

2-9x≤6(1-x)

2 – 9x ≤6-6x

- 9x + 6x ≤ 6-2

- 3x ≤ 4

x ≥ - 4

Evaluation

Solve the inequalities

1)1+4x - 5 + 2x > x -2

2 7

2)2(x- 3) ≤ 5x

Representing the Solutions of Inequalities on a Number Line and on Graphs.

Example

Represent the solutions (i) x ≥ 4 (ii) x < 3 on number line

(i) x

-1 0 4

(ii)

-1 0 3

Note: When it is greater than, the arrow points to the right and vice versa also when “or equal to” is included, in the inequalities, the circle on top is shaded “o” and the “or equal to” is not included the circle is opened “o”

Graphical Representation

Example

Represent the solutions of the inequalities x > 3 and x ≤ 3 graphically

i) x> 3 ii)

1 2 3 -1 0 1 2 3 4

Note: Dotted line (broken line) is used to represent either< or > and when or equal to is included e.g ≤ or ≥ full line is used.

Evaluation:

Solve the inequality 2x + 6 ≤ 5 (x-3) and represent the solution on a number and graphically.

Combining Inequalities

Examples

x ≥ -3 and x ≤ 4 can be combined together to form a single inequality.

x ≥ -3 is the same as -3 ≤ x

- 3 ≤ x and x ≤ 4

-3 ≤ x ≤ 4

-4 -3 -2 -1 0 1 2 3 4 5

If 3+ x ≤5 and 8 + x 5 what range of values of x satisfies both inequalities

Solution

3 + x ≤ 5 8 + x > 5

x ≤5-3 x > 5 - 8

x≤2 x > -3

or -3 < x

then, -3 < x ≤ 2

-3 -2 -1 0 1 2 3 4

The shaded region satisfies the inequalities.

Note: When combining inequalities the inequalities having the lesser value is charged and there are some inequalities that cannot be combined e.g x< -3 and x > 4.

Note: The lesser value has the < sign, and the greater value has the > sign there are two inequalities that can never meet or be combined.

Evaluation

1.If 3 + x ≤ 5 and 8 + x ≥ 5,what range of values of x satisfies both inequalities?

2.State the range of values of x represented by each number line in the figure below.

(a) (b) (c)

-7 -2 -1 0 3 0 -1 4

GENERAL EVALUATION/REVISION QUESTIONS

1.Solve the inequality and sketch a number line graph for its solution

5x-3 – 1-2x ≤ 8 + x

2.If 3 + x ≤5 and 8 + x≥5,what range of values of x satisfies both inequalities?

3.On a Cartesian plane,sketch the region which represent the set of points for which

x<2 and y≥5

Solve the equation ( 6x-2 )/3=(5-3x)/4
Simplify (2a+b)2-(b-2a)2

WEEKEND ASSIGNMENT

Objectives

1.If x varies over the set of real numbers which of the following is illustrated below

-3 -2 -1 0 1 2 3

(a)-3 > x ≤2 (b) -3 ≤x ≤2 (c) -3 ≤ x < 2 (d) -3∠x < 2

2.Solve the inequalities 3m < 9

(a) m< 3 ( m< 2 (c) 4 > m (d) 2 < m

3.If x is a rational no which of the following is represented on the number line?

-8 -6 -4 -2 0 2 4 6

(a) x: -5 ∠x ∠3) (b) x: -4 x <4) (c) x: - 5 ≤ x < 3) (d) x: -5 < x ≤3)

4.Solve the inequality : 5x + 6 ≥ 3 + 2x (a) x≤ 1 (b) x≥ 1 (c) x≥ -1 (d x≤-1

5.Given that a is an integer,find the three highest values of a which satisfy 2a +5 < 16

(a) 3,4,5 (b) 6,7,8 (c) 1,2,3 (d)8,9,10

Theory

If 6x < 2 – 3x and x -7 < 3x what range of values of x satisfies both inequalities (represent the solution on a number line)?

2.Represent the solution of the inequality graphically

x - (x-3) < 1

3 2

Reading Assignment

New General Mathematics SSS2, page 101,exercise10c, numbers 1-10.

Lesson Notes By Weeks and Term - Senior Secondary School 2