3.5 Rational inequality  (Page 2/6)

1: Decompose both numerator and denominator into linear factors. Find critical points by equating linear factors individually to zero.

2: Mark critical points on a real number line. If n be the numbers of critical points, then real number line is divided into (n+1) sub-intervals.

The question however remains that we should know sign of function in at least one interval. We determine the same by testing function value for an intermediate x-value in any of the sub-intervals. Though it is not a rule, we consider a test point in the right most interval, which extends to positive infinity. This helps us to assign signs in the intervals left to it by alternating signs. Sometimes, it may, however, be easier to evaluate function value at x= 0, 1 or -1, provided they are not the critical points. This has the advantage that calculation of function value is easier. Now, these consideration set up the next step of sign diagram :

3: Test sign of function in a particular interval. Assign alternate signs in adjacent sub-intervals.

For the example case, let us put x=0,

$$\Rightarrow f\left(0\right)=\frac{0-0-2}{0-0-8}=\frac{1}{4}>0$$

Thus, sign of function in the interval between -1 and 2 is positive. The signs of function alternate in adjacent sub-intervals.

We have noted that sign of each linear factor combines to determine the sign of rational function. This fact is reflected as sign alternates in adjacent sub-intervals. However, we need to consider the effect of case in which a linear factor is repeated. If a linear factor evaluates to a positive number in an interval and is repeated, then there is no effect on the sign of function. If a linear factor evaluates to a negative number in an interval and is repeated even times, then there is no effect on the sign of function. The product of negative sign repeated even times yield a positive sign and as such does not affect the sign of function. However, if a linear factor evaluates to a negative number in an interval and is repeated odd times, then sign of function changes. Product of negative sign repeated odd times yield a negative sign and as such sign of function changes.

We conclude that if a linear factor is repeated even times, then sign of function will not alternate about the critical point corresponding to linear factor in question. On the other hand, if a linear factor is repeated odd times, then sign of function will alternate as before. Now, these consideration set up the next step of sign diagram :

4: If a linear factor is repeated even times, then sign of function will not alternate about the critical point corresponding to linear factor in question.

In the example case, the linear factor (x+1) is repeated even times (count both in numerator and denominator). As such, sign of function will not change about critical point “-1”. Thus, sign diagram drawn as above need to be modified as :

We can verify modification due to repeated linear factors by putting x = -2 in the function :