Modulus Inequalities

For a positive number $k$, $|x-a|<k$ means less than $k$ units from $a$. Both bounds must hold. By contrast, $|x-a|>k$ allows either of the two outside regions.

Replace strict signs by $\le$ or $\ge$ when equality is allowed. An open circle excludes an endpoint; a filled circle includes it. “And” means overlap; “or” means either region.

Add 5 to all three parts, then divide all three by 2. For $|2x-5|\ge3$, the answer is instead $x\le1$ or $x\ge4$.

If you multiply or divide an inequality by a negative number, reverse its sign. For example, $-2x<6$ gives $x>-3$.

Solve $|1-2x|\le5$ and $|1-2x|>5$.