sign analysis

What is sign analysis?

Direct answer: Sign analysis uses a sign chart to solve polynomial and rational inequalities. Mark where the expression is zero or undefined, then check its sign on each interval. Include a zero only when equality is allowed and the original expression is defined there.

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. Find the zeros of the polynomial or numerator, and the values that make the denominator zero. Keep restrictions from the original denominator even if a factor cancels.
3. Put the critical numbers in order on a number line; they split the domain into test intervals.
4. Choose a test value inside each interval, or find the sign of each factor and combine the signs.
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 both sides by an expression of unknown sign. A negative multiplier reverses the inequality; a positive one does not.
• 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: 2026-09-08. The sign-chart method and endpoint rules were checked against the cited algebra notes. In the example, −1 is excluded and 2 is included, giving (−1, 2].