Third-degree inequality

The first complete example of sign analysis: a third-degree inequality. It is factorised into three first-degree factors, the cornerstones are found and the sign table is filled in.

Example — Third-degree inequality

Solve $x^3-5x^2+6x\le 0$.

Step 1 — Factorise: $x^3-5x^2+6x = x(x^2-5x+6) = x(x-3)(x-2) \le 0$

Step 2 — Associate the factors with lines: $f_1:\; y = x, \quad f_2:\; y = x-3, \quad f_3:\; y=x-2.$

Step 3 — Cornerstones and graph:

Each factor is a line; the cornerstone is the point where it intersects the $x$-axis.

Step 4 — Sign table:

	$x<0$	$x=0$	$0<x<2$	$x=2$	$2<x<3$	$x=3$	$x>3$
$f_1=x$	$-$	$0$	$+$	$+$	$+$	$+$	$+$
$f_2=x-2$	$-$	$-$	$-$	$0$	$+$	$+$	$+$
$f_3=x-3$	$-$	$-$	$-$	$-$	$-$	$0$	$+$
$f_1\cdot f_2\cdot f_3$	$-$	$0$	$+$	$0$	$-$	$0$	$+$

Remark — The rule of signs

In each interval you count the $-$ signs of the factors; if there is an even number the product is $+$, if odd it is $-$.

Solution: we look for where $\le 0$ (sign $-$ or zero): $\boxed{x \le 0 \;\;\text{or}\;\; 2\le x\le 3}$

Quick check: for $x=-1$: $(-1)(-3)(-4)=-12 \le 0$ ✓. For $x=2{,}5$: $(2{,}5)(0{,}5)(-0{,}5)=-0{,}625\le 0$ ✓. For $x=4$: $(4)(2)(1)=8 > 0$ ✓ (rightly outside the solution).

Links

Topics: Inequalities
Concepts: Cornerstone · Factorisation · Sign analysis · Sign table
Methods: Factorising polynomials · Sign analysis of a product
Skills: Solving inequalities · Factorising