sign analysis

What is sign analysis?

Direct answer: Sign analysis, or the sign chart method, solves polynomial and rational inequalities by splitting the number line at every zero and undefined point, then finding the expression's sign on each interval. Keep the intervals that satisfy the inequality, include numerator zeros when equality is allowed, and always exclude denominator zeros.

How to make a sign chart

1. Move every term to one side so the other side is zero, then factor the expression where possible.
2. List the critical numbers: zeros of the numerator or polynomial, plus values that make a denominator undefined.
3. Put the critical numbers in order on a number line; they split the domain into test intervals.
4. Determine the sign of every factor in each interval and combine the signs to get the whole expression's sign.
5. Select the intervals required by >, <, ≥, or ≤. Include an allowed numerator zero for ≥ or ≤, but never include a denominator zero.

Worked example: solve (x − 2)/(x + 1) ≤ 0

The critical numbers are x = −1, where the expression is undefined, and x = 2, where the numerator is zero.

Because the inequality allows equality, include x = 2. Exclude x = −1 because the denominator is zero. The solution is (−1, 2].

Common sign-analysis mistakes

• Multiplying by a denominator before knowing whether it is positive or negative, which can reverse the inequality incorrectly.
• Including a denominator zero in the answer.
• Forgetting that an even-multiplicity factor does not change sign when crossing its zero.
• Testing only the critical points instead of one point inside every open interval.

Sources and review

• Paul's Online Math Notes — Solving Rational Inequalities
• Official CSCA exam introduction

Last reviewed: 2026-08-03. The method and worked solution were checked against the cited algebra reference. Use the official CSCA source for current exam scope and rules.