Combining Inequalities

When combining inequalities, write the smallest number first, followed by the unknown and finally the largest number and vice-versa. 

Example 7.4.1:

Find the inequalities illustrated in the number line shown below: 

Solution 

Smallest number = -4 

Largest number = 2 

Unknown = x

The shaded circle is on -4 (⇒ use \(\scriptsize \geq \) because it’s counting to the right).

The unshaded circle is on 2 (⇒ use \(\scriptsize < \) because its pointing to the left).

We have	\( \scriptsize x \geq -4 \)	and	\( \scriptsize x < 2 \)
Reverse	\( \scriptsize -4 \leq x \)	and	\( \scriptsize x < 2 \)
Combine	\( \scriptsize -4 \leq x < 2 \)

Example 7.4.2:

Find the inequality illustrated in the following diagrams:

(a)  

(b)  

Solution

(a) Separate the inequalities

\( \scriptsize x \geq -1 \) and \( \scriptsize x \leq -4 \)

reverse \( \scriptsize -1 \leq x \) and \( \scriptsize x \leq -4 \)

combine \( \scriptsize -1 \leq x \leq 4 \)

(b) Separate the inequalities 

\( \scriptsize x > 2 \) and \( \scriptsize x \leq 8 \)

reverse \( \scriptsize 2 < x \) and \( \scriptsize x \leq 8 \)

combine \( \scriptsize 2 < x \leq 8 \)

Evaluation Questions: