Triangle inequality
A standard method from Inequalities and linear programming: the steps and worked examples.
Steps
The sum of any two sides is greater than the third side.
- Write all three inequalities: a + b > c, a + c > b and b + c > a.
- Solve each one for the unknown side.
- Combine them: the unknown side lies between the difference and the sum of the other two sides.
Worked examples
Example 1: Quadratic inequality (NF): solve x² − x − 6 < 0.
(x + 2)(x − 3) < 0, so −2 < x < 3.
Example 2: Triangle inequality: two sides of a triangle are 4 cm and 9 cm. Find the range of the third side x.
4 + 9 > x → x < 13
4 + x > 9 → x > 5
9 + x > 4 → always true
5 < x < 13For the whole topic, read Inequalities and linear programming: key points, common mistakes and how the DSE tests it.