Inequalities • Topic 2 of 4

Quadratic Inequalities

A quadratic inequality like x² − 5x + 6 > 0 is solved by the wavy-curve (sign-scheme) method, the fastest tool in CAT for these. First, write the expression in factored form: (x − 2)(x − 3). The critical points are the roots, 2 and 3. Mark them on the number line and, starting from the rightmost interval where the expression is positive (for a positive leading coefficient), alternate signs +, −, + as you move left across each simple root. For > 0 pick the positive intervals (x < 2 or x > 3); for < 0 pick the negative interval (2 < x < 3). This same method extends to rational inequalities such as (x − 1)/(x − 4) ≤ 0: include roots of the numerator (where equality holds) but always exclude roots of the denominator, since the expression is undefined there. A repeated factor like (x − 1)² does not change sign, so the curve only touches the axis there.

✅ Solved examples

1. Solve x² − 5x + 6 > 0.
Factor: (x − 2)(x − 3) > 0. Roots 2, 3. Positive outside the roots ⇒ x < 2 or x > 3.
2. Solve x² − x − 6 ≤ 0.
Factor: (x − 3)(x + 2) ≤ 0. Roots −2, 3. Negative between the roots ⇒ −2 ≤ x ≤ 3.
3. Solve (x − 1)/(x − 4) ≤ 0.
Critical points 1, 4. Expression ≤ 0 between them; include 1 (numerator zero), exclude 4 ⇒ 1 ≤ x < 4.
4. Solve x² + 4 > 0 and then x² + 4 < 0.
x² + 4 ≥ 4 > 0 for all real x, so x² + 4 > 0 holds for all x; x² + 4 < 0 has no real solution.

✏️ Practice — try these, take hints as needed

1. Solve x² − 7x + 12 < 0.
Factor it.
(x − 3)(x − 4).
Negative between roots.
3 < x < 4
2. Solve x² − 9 ≥ 0.
(x − 3)(x + 3).
Positive outside roots.
Include the roots.
x ≤ −3 or x ≥ 3
3. Solve (x + 2)(x − 5) > 0.
Roots −2 and 5.
Positive outside.
Strict, so exclude roots.
x < −2 or x > 5
4. Solve (x − 2)/(x + 3) ≥ 0.
Critical points 2, −3.
Positive outside the interval.
Exclude −3 (undefined).
x < −3 or x ≥ 2
5. Solve x² + 2x + 5 < 0.
Discriminant = 4 − 20.
No real roots; leading coeff > 0.
Expression always positive.
No real solution

📝 Topic test — 8 questions

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