# Lesson Notes By Weeks and Term - Senior Secondary School 2

LINEAR INEQUALITIES IN ONE VARIABLE

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

1. 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)

3

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

3

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)

x

-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

1. 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

1. 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

1. 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

1. Solve the equation ( 6x-2 )/3=(5-3x)/4
2. 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)  -3x < 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

1. 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