4.2 Solving Modulus Inequalities

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📚 Main Content

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Section 1: Quick Recap — What is a Modulus?

Before solving inequalities, make sure you remember what a modulus (also called absolute value) means.

The modulus of a number is its distance from zero — it is always zero or positive.

• $|5| = 5$
• $|-5| = 5$
• $|0| = 0$

So $|x|$ simply means: "take the positive version of whatever x equals."

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✅ Property 1 — "Less than" (or less than or equal to)

$|p| \leq q \iff -q \leq p \leq q$

In plain English: If the modulus of something is less than $q$, then that something must lie between $-q$ and $q$.

Example: $|x| < 5$ means $-5 < x < 5$

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✅ Property 2 — "Greater than" (or greater than or equal to)

$$|p| \geq q \iff p \leq -q \text{ or } p \geq q$$

In plain English: If the modulus of something is greater than $q$, then that something must be either very negative (less than $-q$) or very positive (greater than $q$). There are two separate regions.

Example: $|x| > 5$ means $x < -5$ or $x > 5$

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✅ Property 3 — The Squaring Property (for modulus vs modulus)

$|p| \geq |q| \iff p^2 \geq q^2$

In plain English: When both sides of an inequality have a modulus sign, you can square both sides to remove the modulus signs. This works because squaring a modulus gives the same result as squaring the expression inside.