Find where it equals zero first: To solve a quadratic inequality, first find the roots (solve = 0). The roots split the number line into regions; the inequality is true in some of them.
IB-style question — get the roots
For the inequality x² − x − 6 > 0, find the roots that mark the regions.
Step by step
- Solve x² − x − 6 = 0.
- Roots.
Final answer
Roots at x = −2 and x = 3 — they divide the line into three regions.
Rearrange to one side first: Always get the inequality into the form (quadratic) > 0 (or < 0, ≥, ≤) with zero on the right before finding roots.
Let the parabola decide the sign: For an upward parabola (a > 0): it is below zero (negative) BETWEEN the roots, and above zero (positive) OUTSIDE the roots. (A downward parabola is the opposite.)
Upward parabola (a > 0)
- f(x) < 0 → between the roots.
- f(x) > 0 → outside (x < p or x > q).
- It dips below the axis in the middle.
Downward parabola (a < 0)
- f(x) > 0 → between the roots.
- f(x) < 0 → outside the roots.
- It rises above the axis in the middle.
Sketch a quick parabola: Draw the U (or ∩), mark the roots, and shade where it's above/below the x-axis — the inequality reads straight off.
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IB-style question — an upward parabola, > 0
Solve x² − x − 6 > 0.
Step by step
- Roots: (x − 3)(x + 2) = 0 ⇒ x = −2, 3.
- a = 1 > 0 (upward); > 0 means OUTSIDE the roots.
Final answer
x < −2 or x > 3.
IB-style question — the ≤ version
Solve x² − x − 6 ≤ 0.
Step by step
- Same roots −2 and 3; ≤ 0 means BETWEEN (and including) the roots.
Final answer
−2 ≤ x ≤ 3.
≤ includes the roots; < does not: Use closed ends (≤, ≥) when the inequality includes equality, and open ends (<, >) when it doesn't.
Two pieces for 'outside', one for 'between': "Outside" the roots needs two inequalities joined by or (x < p or x > q). "Between" is a single chain (p ≤ x ≤ q). Match the connector to the picture.
IB-style question — a downward parabola
Solve 8 − 2x − x² ≥ 0.
Step by step
- Roots of 8 − 2x − x² = 0 (i.e. x² + 2x − 8 = 0).
- Downward parabola; ≥ 0 (positive) is BETWEEN the roots.
Final answer
−4 ≤ x ≤ 2.
Flip carefully if you multiply by −1: If you multiply or divide an inequality by a negative, reverse the inequality sign. Often easier to keep a > 0 by rearranging instead.