Inequality with absolute value

Statement

Solve the inequality: $|x^2 - 4| < 3$

Solution

Recognising the strategy. Inequality of the form $|A| < k$ with $k>0$: it is equivalent to the double inequality $-k < A < k$, with no need to split into two systems. $-3 < x^2 - 4 < 3$ Adding $4$ to all sides: $1 < x^2 < 7$ The condition $x^2 > 1$ gives $x < -1$ or $x > 1$; the condition $x^2 < 7$ gives $-\sqrt{7} < x < \sqrt{7}$. Intersecting: $\boxed{-\sqrt{7} < x < -1 \quad\text{oppure}\quad 1 < x < \sqrt{7}}$

Links

Topics: Unifying methods
Concepts: Absolute value
Skills: Reasoning by cases · Solving inequalities
Exercise type: Solving an inequality