4.1 Solving Modulus Equations

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What is a Modulus (Absolute Value)?

Before solving any equations, you need to understand what the modulus symbol means.

The modulus of a number is its distance from zero on a number line. Distance is always positive, so the modulus of any number is always zero or positive — never negative.

The modulus is written using two vertical bars: |x|

Examples:

• |5| = 5 (5 is already positive, so it stays as 5)
• |−5| = 5 (−5 is negative, so we make it positive)
• |0| = 0

The formal rule:

$$|x| = x \quad \text{if } x \geq 0$$ $$|x| = -x \quad \text{if } x < 0$$

This means: if the expression inside the bars is positive or zero, keep it as it is. If it is negative, multiply it by −1 to make it positive.

Example:

• |3x − 1| = 3x − 1 when 3x − 1 ≥ 0 (i.e. when x ≥ 1/3)
• |3x − 1| = −(3x − 1) when 3x − 1 < 0 (i.e. when x < 1/3)

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The Graph of y = |ax + b|

The graph of a modulus function looks like a V-shape.

• The graph of y = |x| is a V with its vertex (tip/bottom point) at the origin (0, 0)
• The graph of y = |ax + b| has its vertex where ax + b = 0, i.e. at x = −b/a
• The arms of the V always go upward — the y-values are never negative

Example: y = |x − 5|

• Vertex is where x − 5 = 0, so at x = 5 → vertex is at (5, 0)
• When x > 5: y = x − 5 (a straight line going up to the right)
• When x < 5: y = −(x − 5) = −x + 5 (a straight line going up to the left)

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Type 1: Solving |ax + b| = c

This is the simplest type. Here, c is a constant (a fixed number) and c ≥ 0.

The key idea: If |something| = c, then that "something" equals +c or −c.

$$|ax + b| = c \implies ax + b = c \quad \text{or} \quad ax + b = -c$$