Graphs of Linear Inequalities

Inequalities are said to be linear if they have no square or higher powers of the unknown. In other words, the highest power of the unknown is 1.

For example:

\( \scriptsize 2x > 10 \)

\( \scriptsize 4x \: – \: 5y > \: -18 \)

Number Line:

A number line is a visual representation of a set of real numbers as a series of points.

Showing Inequalities on a Number Line: 

On a number line, you can show inequality by obeying these rules. 

For ‘<’ the arrow points to the left, but the starting point is not shaded. 

For Example, x < 5 is drawn this way

For ‘>’, the arrow points to the right and the starting point is not shaded.

E.g. x > 5 is drawn this way 

For \( \scriptsize ‘ \leq ‘ \), the arrow points to the left and the starting point is shaded.

e.g. \( \scriptsize x \leq \: -2 \), is drawn this way. 

For \( \scriptsize ‘ \geq ‘ \) , the arrow points to the right and the starting point is shaded.

e.g \( \scriptsize x \geq \: -2 \), is drawn this way. 

Example 7.3.1: 

Show the following inequalities on a number line.

(a) \( \scriptsize x > 3 \)

(b) \( \scriptsize x \leq 3 \)

(c) \( \scriptsize x \geq 3 \)

Solution

(a) \( \scriptsize x > 3 \)

(b) \( \scriptsize x \leq 3 \)

(c) \( \scriptsize x \geq 3 \)

Example 7.3.2:

Show each of these inequalities on a number line 

(a) \( \scriptsize x < 4 \)

(b) \( \scriptsize x > 4 \)

(c) \( \scriptsize x \leq 4 \)

(d) \( \scriptsize x \geq 4 \)

Solution

(a) \( \scriptsize x < 4 \)

(b) \( \scriptsize x > 4 \)

(c) \( \scriptsize x \leq 4 \)

(d) \( \scriptsize x \geq 4 \)

Example 7.3.3:

Use suitable symbols to write the inequalities shown in the following number line

(a)

(b)

(c)

(d)

Answers

(a) \( \scriptsize x \geq -3 \)

(b) \( \scriptsize x \leq 2 \)

(c) \( \scriptsize x \geq -1 \)

(d) \( \scriptsize x < 5 \)