Solving Rational Inequalities

Pre-Calculus · Axiom Academy

LESSON Solving Rational Inequalities Get zero on one side, find the critical values, and let a sign chart on the number line pick the answer. 1. Why You Cannot Cross-Multiply Take . It is tempting to multiply both sides by x-1 . But x-1 is positive when and negative when — and multiplying an inequality by a negative number flips it. Since we don't know the sign in advance, that step is invalid. Don't multiply — instead move everything to one side Combine over a common denominator, compared to 0 2. Critical Values Split the Number Line Once the inequality reads , the only places its sign can change are the critical values . There are exactly two kinds, and they behave differently at the endpoints. . The expression equals 0 here — allowed for , so this endpoint can be closed . . The expression is undefined here, so this endpoint is always open — never included. 3. Test the Sign in Each Region Pick a convenient test value in each of the three regions and track the sign of each factor. The sign of is the sign of (x-3) divided by the sign of (x-1) . Two negatives make a positive on the left; the lone sign flip at x=1 makes the middle negative; both factors positive on the right. We want , so the positive regions win. 4. Select the Solution and Mark the Endpoints For we keep the regions where the expression is positive, then decide each endpoint: x=3 makes it 0 (allowed by , so closed ), while x=1 makes it undefined ( open ).

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