For d 0, |q(x)| <= d means -d <= q(x) <= d, an intersection. In contrast |q(x)|>d means q(x)<-d or q(x)>d, a union. Solve each quadratic bound using its roots and sign, then combine the sets using the required AND or OR. Check whether a square bound is automatic or impossible. |x2-2| 1 becomes 1 <= x2 3, giving [- 3,-1] [1, 3]. Check: if d<0, an absolute value is always greater than d and never at most d. At d=0, distinguish roots from all other inputs.

For \(d\ge0\), \(|q(x)|\le d\) means \(-d\le q(x)\le d\), an intersection. In contrast \(|q(x)|>d\) means \(q(x)<-d\) or \(q(x)>d\), a union.

Solve each quadratic bound using its roots and sign, then combine the sets using the required AND or OR. Check whether a square bound is automatic or impossible.

\(|x^2-2|\le1\) becomes \(1\le x^2\le3\), giving \([-\sqrt3,-1]\cup[1,\sqrt3]\).

Check: if \(d<0\), an absolute value is always greater than \(d\) and never at most \(d\). At \(d=0\), distinguish roots from all other inputs.

Quadratic modulus inequalities

Quadratic modulus inequalities