Unit 13 Section 1 : Linear Inequalities

In this section we look at how to solve linear inequalities and illustrate their solutions using a number line.

When using a number line, a small solid circle is used for ≤ or ≥ and a hollow circle is used for > or <.

For example,

x ≥ 5

Here the solid circle means that the value 5 is included.

x < 7

Here the hollow circle means that the value 7 is not included.

When solving linear inequalities we use the same techniques as those used for solving linear equations. The important exception to this is that when multiplying or dividing by a negative number, you must reverse the direction of the inequality. However, in practice, it is best to try to avoid doing this.

Example 1

Solve the inequality x + 6 > 3 and illustrate the solution on a number line.

x + 6 > 3
x > 3 – 6Subtracting 6 from both sides of the inequality
x > –3

This can be illustrated as shown below:

Example 2

Solve the inequality 3x + 7 ≥ 19 and illustrate the solution on a number line.

3x + 7 ≥ 19
3x ≥ 12Subtracting 7 from both sides
x ≥ 4Dividing both sides by 3

This can now be shown on a number line.

Example 3

Illustrate the solution to the inequality 12 – 3x ≥ 6.

Because this inequality contains the term ' –3x ', first add 3x to both sides to remove the – sign.

12 – 3x ≥ 6
12 ≥ 6 + 3xAdding 3x to both sides
6 ≥ 3xSubtracting 6 from both sides
2 ≥ xDividing both sides by 3
or   x ≤ 2

This is illustrated below.

Example 4

Solve the equation –7 < 5x + 3 ≤ 23.

In an inequality of this type you must apply the same operation to each of the 3 parts.

–7 < 5x + 3 ≤ 23
–10 < 5x ≤ 20Subtracting 3 from both sides
–2 < x ≤ 4Dividing both sides by 5

This can then be illustrated as below.

Exercises

Question 1

Draw diagrams to illustrate the following inequalities:

Note: on the number line click more to change the ends of the lines.

(a)
x > 3
(b)
x ≤ 4
(c)
x ≤ –2
(d)
x ≥ –3
(e)
–2 ≤ x < 4
(f)
0 ≤ x ≤ 3
Question 2

Write down the inequality represented by each of the following diagrams:

(a)
x
(b)
x
(c)
x
(d)
x
(e)
x
Question 3

Solve each of the following inequalities and illustrate the results on a number line.

(a)

x + 7 > 12

x
(b)

x – 6 > 3

x
(c)

4x ≤ 20

x
(d)

5x ≥ 10

x
(e)

x + 6 ≥ 8

x
(f)

x – 6 ≤ –3

x
(g)

x + 8 ≤ 5

x
(h)

≥ 3

x
(i)

≤ –1

x
Question 4

Solve each of the following inequalities and illustrate your solutions on a number line.

(a)

6x + 2 ≥ 8

x
(b)

5x – 3 > 7

x
(c)

3x – 9 < 6

x
(d)

4x + 2 ≤ 30

x
(e)

5x + 9 ≤ –1

x
(f)

4x + 12 > 4

x
(g)

+ 4 > 3

x
(h)

– 1 ≤ –3

x
(i)

+ 6 ≤ 5

x
Question 5

Solve the following inequalities, illustrating your solutions on a number line.

(a)

–1 ≤ 3x + 2 ≤ 17

x
(b)

4 – 6x < 22

x
(c)

5 – 3x ≥ –1

x
(d)

14 ≤ 4x – 2 ≤ 18

x
(e)

20 – 8x < 4

x
(f)

32 – 9x ≥ –4

x
(g)

11 – 3x ≤ 20

x
(h)

–11 ≤ 3x – 2 ≤ –5

x
(i)

–7 < 2x + 5 ≤ 1

x
Question 6

Given that the perimeter of the rectangle shown is less than 44, form and solve an inequality.

x
(x + 8) + x + (x + 8) + x < 44
4x + 16 < 44
4x < 28
x < 7
Question 7

The perimeter of the triangle shown is greater than 21 but less than or equal to 30.

Form and solve an inequality using this information.

x
21 < (x + 1) + (x + 2) + (x + 3) ≤ 30
21 < 3x + 6 ≤ 30
15 < 3x ≤ 24
5 < x ≤ 8
Question 8

The area of the rectangle shown must be less than 50 but greater than or equal to 10.

Form and solve an inequality for x.

x
Question 9

A cyclist travels at a constant speed v miles per hour. He travels 30 miles in a time that is greater than 3 hours but less than 5 hours.

Form an inequality for v.

v
Question 10

The area of a circle must be greater than or equal to 10 m² and less than 20 m² . Determine an inequality that the radius, r, of the circle must satisfy.

m <r< m   (to the nearest cm)
10 ≤ πr² < 20
≤ r <
1.784124116 m ≤ r < 2.523132522 m
1.78 m < r < 2.52 m   (to the nearest cm)
Question 11
R
R
R
R
R
R
R
R
R

The pattern shown is formed by straight lines of equations in the first quadrant.

(a)

One region of the pattern can be described by the inequalities

x ≤ 2
x ≥ 1
y ≥ x
y ≤ 3

Click to put an R in the single region of the pattern that is described.

This is another pattern formed by straight line graphs of equations in the first quadrant.

(b)

The shaded region can be described by three inequalities.

Write down these three inequalities.

y <
y >
x <