Quadratic inequalities like x squared minus 5x + 6 > 0 are solved with a sign chart: find the zeros, split the number line into intervals, test one value in each interval, and pick the regions where the inequality holds. The ACT rarely asks you to graph the parabola, but the sign-chart method is fast and reliable.
Step 1: Find the zeros
Factor or use the quadratic formula. For x squared minus 5x + 6 > 0, factor to (x - 2)(x - 3) > 0. The zeros are x = 2 and x = 3. Those are the only places the expression can change sign. Factoring speed comes from factoring quadratics on the ACT.
Step 2: Build the sign chart
Mark 2 and 3 on a number line. That creates three intervals: x < 2, 2 < x < 3, and x > 3. Pick a test value in each interval and check whether the factored form is positive or negative.
- x = 0 (left interval): (-)(-) = positive
- x = 2.5 (middle): (+)(-) = negative
- x = 4 (right): (+)(+) = positive
For > 0, we want positive intervals: x < 2 or x > 3. For ≥ 0, include the zeros where the expression equals zero.
When factoring fails
If the quadratic does not factor cleanly, use the quadratic formula to find the zeros, then run the same sign chart. Review quadratic equations on the ACT for the formula and discriminant. The sign-chart logic does not change.
Parabola intuition
If the leading coefficient is positive, the parabola opens up—the expression is negative between the roots and positive outside. If it opens down, flip that. A quick sketch confirms your chart and catches flipped-interval mistakes.
Linear vs quadratic
Linear inequalities flip the sign when you multiply by a negative—see inequalities on the ACT. Quadratic inequalities do not use that flip; they use zeros and interval tests. Mixing the two methods is the main mistake. Know which type you are looking at before you start.
Start practicing
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This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.