Many problems in mathematics, science, and engineering require us to describe quantities that lie within a certain distance of a target value, or beyond a certain distance from a reference point. For example:

• A manufactured component must be within $0.02\,\text{mm}$ of its specified diameter.
• A chemical reaction requires the temperature to stay within $5°\text{C}$ of $200°\text{C}$.
• In calculus, the formal ($\varepsilon$-$\delta$) definition of a limit: $\lim_{x \to a} f(x) = L$ means for every $\varepsilon > 0$ there exists $\delta > 0$ such that $0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$.

All of these situations are naturally expressed using inequalities with modulus (absolute value). This lesson will equip you with the definitions, theorems, techniques, and intuition needed to solve such inequalities confidently.