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
- First swap the > for an = and solve the matching equation. Factorise the left side.
- Each bracket = 0 gives a root. These two x-values are the ONLY places the curve crosses the x-axis — so the only places it can switch between positive and negative.
- Mark both roots on the number line. Two marks chop the line into THREE stretches (just like two snips in a ribbon give three pieces): left of −2, between −2 and 3, and right of 3.
Final answer
The roots are x = −2 and x = 3 — the two points where the curve meets the x-axis. They split the number line into THREE regions; the inequality is true in some of them (we test which next).
Animated graph
Watch the graph build step by step in study mode.
Rearrange to one side first: Always get the inequality into the form (quadratic) > 0 (or < 0, ≥, ≤) with zero on the right before finding roots.
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The shape tells you the sign — no testing needed: Once you have the roots, just picture the parabola.
An upward parabola (a > 0, a U-shape) dips below the x-axis between the roots and sits above it outside them. So:
• below zero (< 0) → BETWEEN the roots
• above zero (> 0) → OUTSIDE the roots
A downward parabola (a < 0, a ∩-shape) is the mirror image: above between, below outside.
Animated graph
Watch the graph build step by step in study mode.
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.
Animated graph
Watch the graph build step by step in study mode.
≤ includes the roots; < does not: Use closed ends (≤, ≥) when the inequality includes equality, and open ends (<, >) when it doesn't.
Two pieces or one? Match the picture: Look at which part of the axis your sketch shades:
• Shaded the two outside arms → write two inequalities joined by or: x < p or x > q.
• Shaded the single middle stretch → write one chain: p ≤ x ≤ q.
That's the whole choice — 'outside' uses or, 'between' is one chain.
IB-style question — a downward parabola
Solve 8 − 2x − x² ≥ 0.
Step by step
- Step 1 — find the roots. To find them, solve the matching equation 8 − 2x − x² = 0. The leading −x² is awkward to factorise, so multiply every term by −1. The right side is 0 and 0 × −1 = 0, so this is totally safe — there's no inequality to flip yet, we're just solving = 0.
- Now it factorises easily — two numbers that multiply to −8 and add to +2 are +4 and −2.
- Step 2 — which way does it open? Read the ORIGINAL 8 − 2x − x²: the x² term is −x², so a = −1 < 0. The parabola opens DOWN — a ∩ arch.
- Step 3 — apply the shape. We want ≥ 0, i.e. ON or ABOVE the axis. A ∩ arch is above the axis BETWEEN its roots, so take that middle stretch with the ends included.
Final answer
−4 ≤ x ≤ 2 — the arch sits on or above the x-axis between its two roots.
Animated graph
Watch the graph build step by step in study mode.
Flip carefully if you multiply by −1: Multiplying an equation (= 0) by −1 is always safe. But if you ever multiply or divide an inequality by a negative, you must reverse the inequality sign. Keeping a > 0 by rearranging avoids the trap.