A7LAM CSCA ACADEMYCSCA Academy / Mathematics / Sets and inequalities
CSCA Mathematics · Lesson 3 of 16Inequalities
Inéquations · المتراجحات
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Inequalities work like equations with one trap: multiplying or dividing by a negative number reverses the sign. Quadratic and rational inequalities are solved with the roots and a sign check.
Key ideas
- Multiply or divide by a negative number: reverse the inequality sign.
- ax² + bx + c < 0 with a > 0: the solution lies between the roots.
- ax² + bx + c > 0 with a > 0: the solution lies outside the roots.
- f(x)/g(x) > 0: f and g have the same sign, and g(x) ≠ 0.
Formulas to know
- |x| < a ⇔ −a < x < a
- |x| > a ⇔ x < −a or x > a
- a² + b² ≥ 2ab (equality when a = b)
Worked example
Solve x² − x − 6 < 0.
- Roots: (x + 2)(x − 3) = 0 → x = −2 or x = 3
- Sign: a = 1 > 0, the parabola is below 0 between the roots
- Answer: −2 < x < 3
Answer: −2 < x < 3
Practice questions
1. Solve −2x + 4 > 10.
x > −3 · x < −3 ✓ · x > 3 · x < 7
−2x > 6; dividing by −2 reverses the sign: x < −3.
2. Solve |x − 1| ≤ 2.
−1 ≤ x ≤ 3 ✓ · x ≤ 3 · 1 ≤ x ≤ 3 · x ≤ −1 or x ≥ 3
−2 ≤ x − 1 ≤ 2, so −1 ≤ x ≤ 3.
3. Solve (x − 1)/(x + 2) > 0.
−2 < x < 1 · x > 1 · x < −2 or x > 1 ✓ · x ≠ −2
Both factors positive (x > 1) or both negative (x < −2).