Rational Inequalities (Leaving Cert Mathematics): Quizzes

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Rational Inequalities
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What form do rational inequalities take?

ax+bcx+d\frac{ax+b}{cx+d}

Why can't we multiply x−8x+2≥−1\frac{x-8}{x+2} \ge -1 by (x+2)(x+2) directly?

We don't know if (x+2)(x+2) is positive/negative

What should we multiply by instead of (x+2)(x+2)?

(x+2)2(x+2)^2

Why is multiplying by (x+2)2(x+2)^2 helpful?

It's guaranteed positive

What can be cancelled from (x−8)(x+2)(x+2)x+2\frac{(x-8)(x+2)(x+2)}{x+2}?

(x+2)(x+2)

After expanding and simplifying, what inequality results?

2x2−2x−12≥02x^2-2x-12 \ge 0

What method finds the roots of 2x2−2x−12≥02x^2-2x-12 \ge 0?

Quadratic formula

What are the roots of 2x2−2x−12=02x^2-2x-12 = 0?

x=3x=3 and x=−2x=-2

After finding roots x=3x=3 and x=−2x=-2, what must we do?

Evaluate region test

What guarantees we don't switch the inequality sign when multiplying?

Multiplying by a positive expression

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Inequality Proofs & Discriminants

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