Systems of second-degree inequalities

A solution of a system of inequalities is any x satisfying all the inequalities at once. So we solve each inequality separately (a quadratic one by the graph or the interval method), mark the solutions on one number line and take their common part. Inequalities with absolute value reduce to systems: for a > 0, |f(x)| < a is equivalent to −a < f(x) < a, i.e. a system of two inequalities; |f(x)| > a is equivalent to f(x) < −a or f(x) > a. Finding a domain also gives a system: each expression under a square root must be non-negative, and a denominator must not be zero. Write the answer from the common part on the number line; if there is no common part, the system has no solution.