Sketch graphs involving the modulus (absolute value) function and solve equations and inequalities containing modulus expressions.

What this dot point is asking

You need to sketch graphs involving the modulus function and to solve equations and inequalities that contain modulus expressions.

Definition and key properties

Sketching modulus graphs

Two distinct transformations get confused, so keep them separate:

• $y = |f(x)|$: sketch $y = f(x)$, then reflect every part that is below the $x$-axis up to above it. Points already at or above the axis stay put. The result never dips below the $x$-axis and typically has sharp corners where $f$ crossed the axis.
• $y = f(|x|)$: keep the graph for $x \ge 0$, then reflect it in the $y$-axis to produce the $x < 0$ side. The result is symmetric about the $y$-axis and ignores the original left half.

Solving modulus equations

The cleanest general method is to split at the points where each modulus argument changes sign, solve on each interval, and discard solutions that fall outside the interval assumed.

For equations of the form $|A| = |B|$, squaring is often faster, since $|A| = |B| \iff A^2 = B^2 \iff A = \pm B$.

Solving modulus inequalities

Two standard equivalences handle most cases, for $c > 0$:

For $|f(x)| < c$, replace $x$ by $f(x)$ and solve the resulting double inequality. A reliable alternative is to square (valid because both sides are non-negative) and solve the polynomial inequality, then sketch to read off the solution set.