MODULE 9

LINEAR INEQUALITIES

Notes

1. An inequality is a relationship

between two unequal quantities.

2.

Symbol Meaning

> Greater than

< Less than

Greater than or equal to

Less than or equal to

3. Representation of an inequality on a

number line

(a)

x 2

-3 -2 -1 0 1

The empty circle ‘o’ indicates that the

values of x do not include -2

(b)

x 2

- 1 0 1 2 3

The solid circle ‘’ indicates that the

values of x include 2

4. To solve a linear inequality in one

unknown is to find its equivalent

inequalities in the simplest form.

x 1 3

x 3 4

x 7

5. The symbol of inequality is reversed

when both sides of an inequality is

multiplied or divided by a negative

number

(a)

2x 10

x 10

2

x 5

6. Solution for two simultaneous linear

inequalities in one unknown is to find

the common values or equivalent

inequalities that satisfy both the

inequalities.

(a)

x 2

x 0

The common solution for

x 2 and

x 0 is

x 0

(b)

x

3 5

x 5 (3)

x 15

-3 -2 -1 0 1

(b)

x 2

x 0

The common solution for

x 2 and

x 0 is

2 x 0

(c)

x 0

x 2

The common solution for

x 0 and

x 2 is

x 2

d.

x 2

x 0

No common values hence no solution

for

x 2 and

x 0.

EXERCISE

i. Fill in the boxes with the symbol “<”

or “>”

a 39 ⓪ 42

b 17 ⓪ -8

c -23 ⓪ -1

d -80 ⓪ 0

e 20 + 8 ⓪ 5 + 9

f 2 ⓪ 4

g 3 ⓪ -9

h 0 ⓪ -12

ii. Represent each of the following

inequalities on a number line

a

x 8

-3 -2 -1 0 1

-3 -2 -1 0 1

-3 -2 -1 0 1

b

x 12

c

x 2

d

35 y

iii. Write a linear inequality in one

unknown for each of the following.

a

42

b

100

c

85

d

-96

iv. Solve

a

x 3 4

b

x 1729

c

y (2) 13

d

y 315

e

p 7 3

f

3 q 12

g

4x 3 3x 9

h

1 5y 14 4y

v. Solve

a

4x 20

b

3

4y 9

c 7p < 49

d

y

5 1

e 5y < 25

f

8m 64

g 6b > 54

h

105x

i

18 3x

j

15 11y

vi Solve

a

4x 20

b

3

4y 9

c -7p < 49

d

y

5 1

e -5y < 25

f

m 64

g -6b > 54

h

105x

i

183x

j

15 11y

vii. Solve

a 1 – y < 3

b

3 x 11

c

4a1959

d

53x 11

e

1 2(x 3)

f

x

2 8 10

g

5 x 63x

h

5(x 1) x 8

i

x 2

3 1

j

p

3p

4 1

k

12y 3

y 4

l

3b1

b 2

m

1 5x

2 6

n

3(4 x) 36

viii Represent the common values of

each pair of simultaneous linear

inequalities on a number line.

a x > -2 and x > -6

b

x 3 and x < -13

c

x 4 and

x 4

ix Determine the equivalent

inequalities for the following pairs of

inequalities.

a x > 1 and x > 14

b x < -5 and

x 13

c

x 3.5 and x < 10

d

x 7and

x 11

x. Solve the following simultaneous

linear inequalities.

a

x

3 2 and x – 1 > 10

b

1 x 3 and

2x 18

c

3x 513 and x < -2