Compound inequalities

A compound inequality consists of two inequalities combined into a single condition. Instead of only saying that an unknown is greater or smaller than one number, we can restrict it to lie between two bounds.

 

Example

A classic compound inequality could look like this:

 

$$ \large 1 < x < 5 $$

 

This means that \( \large x \) must be greater than 1 and at the same time less than 5. The solution is therefore all numbers between 1 and 5.

 

 

Two types of compound inequalities

And-inequalities (intersection): where both conditions must be satisfied at the same time. For example:

 

$$ \large x > 0 \; \text{and} \; x < 10 $$

 

Or-inequalities (union): where it is enough to satisfy one of the conditions. For example:

 

$$ \large x < -2 \; \text{or} \; x > 3 $$

 

 

Solving compound inequalities

To solve a compound inequality, you work step by step with both sides in the same way as with a regular inequality.

 

Example:

 

$$ \large 2 < 3x + 1 < 8 $$

 

We subtract 1 everywhere:

 

$$ \large 2 - 1 < 3x < 8 - 1 $$

$$ \large 1 < 3x < 7 $$

 

Divide by 3 everywhere:

 

$$ \large \frac{1}{3} < x < \frac{7}{3} $$

 

The solution is therefore all numbers between \( \large \frac{1}{3} \) and \( \large \frac{7}{3} \).

 

 

Example with "or"

If we have an inequality like:

 

$$ \large x \leq -4 \quad \text{or} \quad x > 2 $$

 

This means that the solution consists of two intervals: all numbers less than or equal to -4, or all numbers greater than 2. Here it is not continuous, but two separate intervals.

 

 

Summary

• Compound inequalities combine two inequalities into a single condition.
• They can either be "and"-inequalities (both conditions must be satisfied) or "or"-inequalities (it is enough to satisfy one of them).
• They are solved by working with both sides in the same way as with a regular inequality.
• The solution is often an interval – or several intervals.