|u| <= v requires v 0 and is equivalent to -v <= u <= v. The allowed height cannot be negative. Write the right-side condition, solve the two linear bounds and intersect all conditions. Squaring is safe only after ensuring the right side is nonnegative. For |x-1| 2x, require x 0. The bounds give x >= 13 and x >= -1, hence x >= 13. Check: joining the two bounds with OR gives a wrong answer. A negative right side cannot be at least a nonnegative absolute value.

\(|u|\le v\) requires \(v\ge0\) and is equivalent to \(-v\le u\le v\). The allowed height cannot be negative.

Write the right-side condition, solve the two linear bounds and intersect all conditions. Squaring is safe only after ensuring the right side is nonnegative.

For \(|x-1|\le2x\), require \(x\ge0\). The bounds give \(x\ge\dfrac{1}{3}\) and \(x\ge-1\), hence \(x\ge\dfrac{1}{3}\).

Check: joining the two bounds with OR gives a wrong answer. A negative right side cannot be at least a nonnegative absolute value.

Modulus inequalities with a linear right side

Modulus inequalities with a linear right side