Quadratic and Fractional-Rational Inequalities

Quadratic Inequality

ax² + bx + c > 0 (or <, ≥, ≤). Solution method: first find the roots, then take into account the direction of the parabola's branches (the sign of a).

Case (a > 0, D > 0)	Solution
ax² + bx + c > 0	outside the roots: x < x₁ or x > x₂
ax² + bx + c < 0	between the roots: x₁ < x < x₂

• D < 0 and a > 0: the expression is always positive. Therefore the solution of > 0 is the whole number line, and < 0 has no solution.
• D = 0: the expression equals a(x − x₁)², and the sign does not change when passing through the point x₁.
• If a < 0, multiply both sides by −1 and reverse the inequality sign.

Method of Intervals

1. Set the expression equal to zero and mark all the roots on the number line
2. Determine the sign in the rightmost interval
3. The sign changes when passing through a root
4. Write down the intervals with the required sign

• In a strict inequality (<, >), the roots are not included in the solution and are written with round brackets.
• In a non-strict inequality (≤, ≥), the roots are included and are written with square brackets.
• The sign does not change when passing through a root of even multiplicity.

Fractional-Rational Inequality

It has the form f(x)/g(x) ≥ 0. The roots of the numerator and denominator are used in the method of intervals.

• The zeros of the denominator are always excluded.
• The sign of a fraction is the same as the sign of the product, but the condition g(x) ≠ 0 is added.
• Do not multiply through by the denominator, because the sign of the denominator is unknown.
• First the inequality must be brought to the form f(x)/g(x) > 0 (or <, ≥, ≤) with 0 on the right side: bring everything to a common denominator to obtain a single fraction.

Examples

• x² − x − 6 < 0: the roots are −2 and 3, the solution is −2 < x < 3, that is, (−2; 3).
• (x − 1)/(x + 2) ≥ 0: the solution is x < −2 or x ≥ 1, that is, (−∞; −2) ∪ [1; +∞).