Solving Linear Inequalities
Inequalities on a Number Line

An inequality involves one of the four symbols

>, ≥ , < or ≤

The following statements illustrate the meaning of each of them.

x > 1 : x is greater than 1
x ≥ −2 : x is greater than or equal to −2
x < 10 : x is less than 10
x ≤ 12 : x is less than or equal to 12

Inequalities can be represented on a number line, as shown in the following worked examples.

Worked Examples

1

Represent the following inequalities on a number line.

(a)

x ≥ 2

Show me

The inequality, x ≥ 2, states that x must be greater than or equal to 2. This is represented as shown.

Note that solid mark, •, is used at 2 to show that this value is included.
(b)

x < −1

Show me

The inequality x < −1 states that x must be less than −1. This is represented as shown.

Note that a hollow mark, o , is used at −1 to show that this value is not included.
(c)

−2 < x ≤ 4

Show me

The inequality −2 < x ≤ 4 states that x is greater than −2 and less than or equal to 4. This is represented as shown.

Note that o is used at −2 because this value is not included and • is used at 4 because this value is included.
2

Write an inequality to describe the region represented on each number line below.

(a)
Show me

The diagram indicates that the value of x must be less than or equal to 3, which would be written as

x ≤ 3
(b)
Show me

The diagram indicates that x must be greater than or equal to −1 and less than 3. This is written as

−1 ≤ x < 3

Exercises

Represent each of the inequalities below on a number line.

(a)

x > 3

(b)

x < 4

(c)

x > −1

(d)

x < 2

(e)

x ≥ 6

(f)

x ≥ −4

(g)

x ≤ 3

(h)

x ≤ 1

(i)

2 ≤ x ≤ 4

(j)

−1 < x ≤ 2

(k)

−2 < x < 2

(l)

1 ≤ x ≤ 3

Write down the inequality which describes the region shown in each diagram.

(a)
x
(b)
x
(c)
x
(d)
x
(e)
x
(f)
x
(g)
x
(h)
x
(i)
x
(j)
x

The speed limits on a road require drivers to travel at a minimum speed of 30 km/hr and a maximum speed of 70 km/hr.

(a)

Copy the diagram below and represent this information on it.
The letter, V, is used to represent the speed.

(b)

Write down an inequality to describe your diagram.

x

List all the possible integer values of n such that

−3 ≤ n < 2

List the numbers in increasing order and separate with commas.

n: