3.5 Rational inequality

Rational inequality is an inequality involving rational expression. There are four forms of inequality. Corresponding to each of these forms, there are four rational inequality forms. These inequality forms essentially compare a rational expression, f(x), with zero. The four inequalities are :

$$f\left(x\right)<0$$ $$f\left(x\right)\le 0$$ $$f\left(x\right)>0$$ $$f\left(x\right)\ge 0$$

We need to note two important aspects of these inequalities. Solution of inequalities, in general, are not discrete values but set of “x” values in the form of interval or union of intervals. Generally, the inequality holds for a continuum of values. Second aspect is about the basic nature of inequality. We know that zero has special significance in real number system. It divides real number system in positive and negative segments. Therefore, solution of these inequalities is about knowing the sign of function values for different intervals in the domain of the function. Corresponding to four inequalities, we need to know intervals in which rational function is (i) negative (ii) non-positive (iii) positive and (iv) non-negative. In the following section, we shall devise a technique to determine sign of rational expression in different internals.

Sign scheme or diagram for rational function

Sign scheme or diagram is representation of sign in different intervals along real number line. This gives a visual idea about the sign of function. Graphically, sign of function changes when graph crosses x-axis. This means that sign of function changes about the zeroes of function i.e. about real roots of a function. However, rational function is ratio of two functions. A change of sign of either numerator or denominator affects sign of rational function.

We consider here only integral rational functions such that expressions in numerator and denominator can be decomposed into linear factors. Equating each of the linear factors, we determine points about which either or both of numerator and denominator functions change sign. We should understand that each of the linear factors is a potential source of sign change as the value of x changes in the domain. This means that each of the points so determined plays a critical role in deciding the sign of function. For this reason, we call these points as “critical points”.

Let us consider an example here :

$$f\left(x\right)=\frac{x^2-x-2}{x^2-3x-8}=\frac{\left(x+1\right)\left(x-2\right)}{\left(x+1\right)\left(x-4\right)}$$

Critical points are -1, 2, -1 and 4. There are two important things to realize here. First, we can not cancel common linear factors as this will result in loosing undefined points and will loose information on sign change. The marking on real number line is as shown here :

Second, the fact that function may change its sign in the domain has an interesting consequence. It can be better understood in terms of function graph, which is essentially a curve. The event of crossing of x-axis by the graph records the event of change of sign. Another change in the sign of graph warrants that curve should cross x-axis again. This corresponds to reversal of sign. It is not possible to change sign of function without crossing x-axis. This means that function will change sign at critical points. Equivalently, we say that sign of function alternates in consecutive sub-intervals. Now, these considerations set up the first two steps of sign diagram :