ACT Math: Solve Rational Inequalities—Handle Sign Changes and Critical Points

A rational inequality is an inequality with a rational expression, like (x+2)/(x-3)>0. Steps: (1) Move everything to one side so you have an inequality with zero on the other. (2) Find critical points: zeros of numerator and denominator. (3) Use sign analysis: test values in each interval to determine where the expression is positive/negative. (4) Select intervals matching the inequality direction. Example: (x+2)/(x-3)>0. Critical points: x=-2 (numerator zero), x=3 (denominator zero). Test intervals: x<-2, -23. Test x=-3: (-1)/(-6)=positive. Test x=0: (2)/(-3)=negative. Test x=5: (7)/(2)=positive. Solution: x<-2 or x>3. Rational inequalities flip signs at critical points; sign analysis determines which intervals satisfy the inequality.

Note: At x=3, the expression is undefined (denominator=0), so x=3 is never included. At x=-2, numerator=0, so the expression=0; include x=-2 only if the inequality is ≥ or ≤.

Mistake 1: Forgetting to check whether critical points should be included. x=denominator (vertical asymptote) is NEVER included. x=numerator zero may be included if ≥ or ≤. Mistake 2: Using test points incorrectly. Test ANY point in the interval (except critical points). Even one test point per interval suffices. Mistake 3: Misinterpreting sign analysis results. If you need (expression)>0 and your test shows negative, that interval is NOT part of the solution. Mistake 4: Forgetting to consider that multiplying/dividing by negative numbers in pre-setup algebra flips inequality signs. Always verify: Did I flip the inequality when I manipulated the expression? Are critical points included or excluded correctly?

Checklist: (1) Move all terms to one side; set up ineq. vs. 0. (2) Find critical points. (3) Test one value per interval. (4) Identify which intervals satisfy the inequality. (5) Write solution, excluding vertical asymptotes, including zeros if applicable.

Problem 1: (x-1)/(x+2)<0. Critical points: x=1, x=-2. Test: x=-3 gives (-4)/(-1)=positive; x=0 gives (-1)/(2)=negative; x=2 gives (1)/(4)=positive. Solution: -25. (Include x=-3, exclude x=5.) Problem 3: 1/(x-2)>1. Rearrange: 1/(x-2)-1>0 → (1-(x-2))/(x-2)>0 → (3-x)/(x-2)>0. Critical points: x=3, x=2. Solution: 2For each, show critical points, test intervals, and solution with proper inclusion/exclusion.

Daily drill: Solve one rational inequality daily. Practice sign analysis and interval notation. Verify answers by testing boundary values.

About 1 rational inequality question per ACT Math section combines algebra, inequality solving, and sign analysis—multiple skills at once. These feel advanced to many students but follow a straightforward method. Correctly solving a rational inequality signals strong algebraic thinking and methodical problem-solving, making these high-value questions where right answers are less common.

Spend 2 days on rational inequalities. Master sign analysis, practice critical point identification, and drill interval notation. By test day, these questions will feel manageable and you'll gain an advantage.