The Modulus Function

The $\text{modulus}$ (or absolute value) $|x|$ is $x$ with any negative sign stripped away: $|5|=5$ and $|-5|=5$. Formally it is defined in two pieces — it leaves non-negative numbers alone and flips negative ones. Best of all, read it as $\text{distance from zero}$, which is always positive or zero.

That distance reading is the key to the whole topic. $|x-3|$ is “the distance between $x$ and 3”, so $|x-3|=5$ means “$x$ is 5 away from 3”, giving $x=8$ or $x=-2$.

The graph of $y=|x|$ is a $\text{V}$: the line $y=x$ for $x\ge 0$, and its reflection for $x<0$. More generally, $y=|f(x)|$ takes the graph of $f$ and \textbf{reflects any part below the x-axis up above it} — because the modulus makes every output non-negative. The parts already above the axis are untouched.

A different graph, $y=f(|x|)$, keeps the right-hand half of $f$ and mirrors it into the left, producing a curve symmetric about the $y$-axis. Do not confuse the two: the modulus is outside in the first, inside in the second.

Because $|A|=b$ means $A$ is a distance $b$ from zero, there are two possibilities: $A=b$ or $A=-b$. So solve modulus equations by splitting into these two cases, then $\text{check each solution}$ back in the original — squaring or careless casework can introduce false answers.

For an equation like $|f(x)|=|g(x)|$, a clean trick is to square both sides (since both are non-negative), turning it into an ordinary equation with no modulus at all.

The distance picture makes inequalities intuitive. $|x|<b$ means “within $b$ of zero”, i.e. $-b<x<b$ — a band. $|x|>b$ means “further than $b$ from zero”, i.e. $x<-b$ or $x>b$ — two outer regions. Apply the same idea to $|x-a|$: “within $b$ of $a$”.