Last Updated Sep 27, 2025

Double Inequalities Simplified Revision Notes for Leaving Cert Mathematics

Revision notes with simplified explanations to understand Double Inequalities quickly and effectively.

Learn about Inequalities for your Leaving Cert Mathematics Exam. This Revision Note includes a summary of Inequalities for easy recall in your Mathematics exam

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Double Inequalities

In some cases we may encounter inequalities with two inequality signs. In such cases, you can solve both inequalities separately.

Example

infoNote

Solve the inequality

$$3x\le 5x+3 \le23$$

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Separate the inequalities :

$$\underbrace{3x\le 5x+3}_{} \le23$$

Solve :

$$\begin{align*} 3x &\le 5x+3 & \\\\ -2x &\le 3 & \text{\footnotesize\textcolor{gray}{(\(-5x\))}} \\\\ x &\ge -\tfrac{3}{2} &\text{\footnotesize\textcolor{gray}{(\( \div (-2) \), flip sign)}} \\ \end{align*}$$

Separate the second inequality :

$$3x\le \underbrace{5x+3 \le23}$$

Solve :

$$\begin{align*} 5x+3 &\le23 & \\\\ 5x &\le20 & \text{\footnotesize\textcolor{gray}{(\(-3\))}} \\\\ x &\le 4 &\text{\footnotesize\textcolor{gray}{(\( \div 5 \))}} \\ \end{align*}$$

We have solved for two regions for which the inequality will be satisfied, $x$ needs to be greater (or equal) to $-\tfrac{3}{2}$ but less than (or equal) to $4$. Combine the two regions :

$$-\tfrac{3}{2} \le x \le 4$$

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