Inequalities

So far we have looked for the values that make an expression equal to zero; now we want to know when it is positive or negative. It is an apparently small change of perspective, but in reality a fundamental one: instead of finding points, we look for intervals of values. The chapter starts from first-degree inequalities (with the dreaded sign-reversal rule), moves on to second-degree inequalities read off the graph of the parabola, and arrives at the most powerful tool of all: the sign table, which allows you to tackle higher-degree inequalities and rational inequalities by studying the sign factor by factor. The chapter closes with systems of inequalities and literal inequalities with a parameter.

In brief — Equations vs. Inequalities: a comparison of the methods

	Equation ($=0$)	Inequality ($\gtrless 0$)
1st degree	Isolate $x$ using the principles	Isolate $x$; reverse the sign if you multiply by a number $<0$
2nd degree	Formula: $x=\frac{-b\pm\sqrt{\Delta}}{2a}$	Sign of the parabola: read off the graph where it lies above/below the $x$-axis
Degree $\ge 3$	Factorise and set each factor $=0$	Factorise and build the sign table
Rational	Factorise, domain conditions, set $N(x)=0$	Factorise, domain conditions, sign table of $\frac{N(x)}{D(x)}$

Sezioni

• First-degree inequalities
• Second-degree inequalities
• Sign analysis: higher-degree and rational
• Systems of inequalities
• Literal inequalities

Esercizi

• All the worked exercises of the chapter