The sign-table method

Inequalities of degree $\ge 3$ and fractional ones are solved with a systematic method: factorise, find the zeros of each factor, build the sign table, read off the result. The sign table is a tool of surprising power: once learnt, any rational inequality becomes manageable.

In brief — Systematic method in 4 steps

To solve an inequality of degree higher than the first (or a fractional one):

1. Bring everything to the left-hand side and reduce to $\frac{N(x)}{D(x)}\gtrless 0$.
2. If it is fractional, do not remove the denominator; put over a common denominator.
3. Factorise the numerator and denominator: $f_1\cdot f_2\cdot\ldots$
4. Study the sign of each factor and fill in the sign table.

Remark — Graphical method for the sign of a factor

To each first-degree factor $f_i = ax+b$ we associate the line $y=ax+b$. Its intersection with the $x$-axis (that is, the solution of $ax+b=0$) is called the cornerstone (or zero). The sign of the factor is read from the graph: $+$ where the line lies above the $x$-axis, $-$ where it lies below.

For a second-degree factor $f_i = ax^2+bx+c$ the same reasoning holds, but with a parabola: the six rules of the previous section apply.

Watch out — Common mistake: "cross-multiplying" in fractional inequalities

It is completely wrong to turn $\dfrac{A}{B} > 0$ into $A\cdot B > 0$ by “multiplying by $B$”! The sign of $B$ is not known. A fractional inequality is handled with the sign table, never by cancelling the denominator.

Links

Topics: Inequalities
Concepts: Cornerstone · Fractional inequality · Factorisation · Sign analysis · Sign table
Skills: Solving inequalities · Factorising