Inequalities and interval notation

Multiplying or dividing an inequality by a negative number reverses its direction. Adding the same expression to both sides preserves it. For a quadratic inequality, first find the roots, split the line into sign intervals and determine the sign in each interval. A graph or a product-sign table can explain which intervals satisfy the question.

Square brackets include an endpoint; round brackets exclude it. Infinity is never an attained endpoint, so it always takes a round bracket. A union joins separate permitted regions. The roots alone do not answer an inequality question: specify the whole set, and substitute a test value in each selected region if the sign is uncertain.

Worked example

Solve −2(x + 1)(x − 3) ≥ 0 and express the solution as an interval.

1. The zeros are x = −1 and x = 3. Divide by −2 and reverse the inequality: (x + 1)(x − 3) ≤ 0.
2. Between the roots, one factor is nonnegative and the other nonpositive, so the product is nonpositive. Outside, the product is positive.
3. Equality is permitted, so include both roots.

Answer: −1 ≤ x ≤ 3, or [−1,3].

Revise first: Solving quadratic equations.