The modulus of \(x\) is written as \(|x|\).

\[ |x| = x \quad \text{if } x \ge 0 \]

\[ |x| = -x \quad \text{if } x < 0 \]

Examples: \[ |5|=5,\qquad |-5|=5,\qquad |0|=0 \]

Modulus means distance.

\[ |x-a| \]

means the distance between \(x\) and \(a\) on the number line.

The graph of \(y=|x|\) is a V-shape with vertex at \((0,0)\).

\[ y = x \quad \text{for } x \ge 0 \]

\[ y = -x \quad \text{for } x < 0 \]

The graph is symmetrical about the \(y\)-axis.

• \(y=|x-a|\): shift right by \(a\)
• \(y=|x|+b\): shift up by \(b\)
• \(y=-|x|\): reflect in the \(x\)-axis
• \(y=k|x|\): stretch parallel to the \(y\)-axis
• \(y=|kx|\): stretch or compress parallel to the \(x\)-axis

Example: \[ y=|x-2|+3 \] has vertex at \((2,3)\).

Example 1:

\[ |x|=4 \]

So: \[ x=4 \quad \text{or} \quad x=-4 \]

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Example 2:

\[ |x-3|=5 \]

So: \[ x-3=5 \quad \text{or} \quad x-3=-5 \]

Therefore: \[ x=8 \quad \text{or} \quad x=-2 \]

Key results:

\[ |x| < a \quad \Longleftrightarrow \quad -a < x < a \]

\[ |x| \le a \quad \Longleftrightarrow \quad -a \le x \le a \]

\[ |x| > a \quad \Longleftrightarrow \quad x < -a \text{ or } x > a \]

\[ |x| \ge a \quad \Longleftrightarrow \quad x \le -a \text{ or } x \ge a \]

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Example 1:

\[ |x-1| < 3 \]

\[ -3 < x-1 < 3 \]

\[ -2 < x < 4 \]

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Example 2:

\[ |2x+1| \ge 5 \]

\[ 2x+1 \ge 5 \quad \text{or} \quad 2x+1 \le -5 \]

\[ x \ge 2 \quad \text{or} \quad x \le -3 \]

Example: Write \( |2x-4| \) in piecewise form.

First find where the inside is zero:

\[ 2x-4=0 \quad \Rightarrow \quad x=2 \]

So:

\[ |2x-4| = 2x-4 \quad \text{for } x \ge 2 \]

\[ |2x-4| = 4-2x \quad \text{for } x < 2 \]

Sketch \(y=|x+1|-2\).

• Start with \(y=|x|\)
• Shift left by 1
• Shift down by 2

The vertex is: \[ (-1,-2) \]