Last Updated Sep 27, 2025

Modulus Inequalities Simplified Revision Notes for Leaving Cert Mathematics

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

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

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Modulus Inequalities

Modulus inequalities involve the absolute value of a linear function of a variable in the form $|ax+b|$.

Example

infoNote

Solve for $x$ in the following inequality :

$$|2x-3|<5$$

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Square both sides :

We can rewrite the inequality as :

$$\begin{align*} (2x-3)^2 &< 5^2 \\\\ 4x^2-12x+9 &< 25 \\\\ 4x^2-12x-16 &< 0 \\\\ (x-4)(x+1)&< 0 \end{align*}$$

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Same as before, let's do the region test, pick two values outside $-1$ and $4$ and a value between.

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$x=0$

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$$\begin{align*} |2(0)-3|&<5 \\ |-3|&<5 \\ 3 &< 5 \end{align*}$$

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$x=5$

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$$\begin{align*} |2(5)-3|&<5 \\ |10-3|&<5 \\ 7 &\not< 5 \end{align*}$$

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$x=-2$

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$$\begin{align*} |2(-2)-3|&<5 \\ |-4-3|&<5 \\ 7 &\not< 5 \end{align*}$$

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$$-1<x<4$$

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