Cambridge O Level Additional Mathematics · Syllabus 4037 · Equations, Inequalities and Graphs
Cubic Inequality
What is Cubic Inequality?
A cubic inequality asks for the values of x at which a cubic curve lies above or below a given horizontal line. The boundary values are the x-coordinates where the curve meets that line, and between consecutive boundaries the curve stays entirely on one side of it, so the solution is a union of intervals. When the comparison value is zero the boundaries are the roots of the cubic, but when it is any other constant the boundaries are different values entirely and must be found by solving the cubic equal to that constant.
This definition is part of the Equations, Inequalities and Graphs chapter in Cambridge O Level Additional Mathematics.
Questions students ask about Cubic Inequality
How do you solve a cubic inequality like \(f(x)\ge d\) from a graph?
Compare the curve with the horizontal line \(y=d\), not with the \(x\)-axis unless \(d=0\). The boundary values are the \(x\)-coordinates where the curve meets that line, found by solving \(f(x)=d\) exactly, and between consecutive boundaries the curve stays on one side of the line, so the answer is a union of intervals. Never read boundaries off a freehand sketch to one decimal place: a sketch shows shape, not position, and the syllabus asks for exact values such as \(1-\sqrt6\).