The idea

The modulus function

What |x| means, how to sketch y = |f(x)|, and how to solve modulus equations and inequalities — including why some "solutions" must be thrown out.

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Pure · Algebra & functions

What |x| means, how to sketch y = |f(x)|, and how to solve modulus equations and inequalities — including why some "solutions" must be thrown out.

The modulus

|x|

is the size of a number — its distance from

0

, always

\ge 0

|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

|5| = 5

and

|-5| = 5

. (The "

-x

" isn't negative — when

x

is negative,

-x

is positive.) Its graph is a V:

y = |x|

y = x

with the part below the axis flipped up.

Draw

y = f(x)

, then reflect every part that is below the $x$ -axis up above it (the modulus can't be negative). The parts already above stay put. So

y = |2x - 4|

is the line

y = 2x - 4

with its negative part flipped, giving a V with its vertex on the

x

-axis at

x = 2

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