Fractional linear inequalities

Linear Inequalities: Variations

Fractional linear inequalities

Just as in the case of equality in the chapter Linear equations with one unknown, inequalities in which quotients of linear expressions #x# occur, can be solved by reduction to linear inequalities. Below a few examples are given.

A fraction in the form #\dfrac{ax+b}{cx+d}#, where #a#, #b#, #c#, and #d# are fixed numbers with at least one of #c# and #d# distinct from zero, is called a fractional linear expression in #x#.

The case in which #c# and #d# are both zero is excluded because in that case the denominator of the fraction would be equal to zero.

The case in which #c=0# corresponds to a linear expression in #x#, namely, #\frac{a}{d}x+\frac{b}{d}#.

We recall that a set of inequalities (with operators like '#\lor#' and '#\land#', which stand for "or" and "and") is equivalent to another set if the two composite inequalities have the same solution set.

Let #\dfrac{ax+b}{cx+d}# be a fractional linear expression in #x#. Inequality #\dfrac{ax+b}{cx+d}\ge0# is equivalent to \[\left(ax+b\ge0\land cx+d\gt0\right)\lor\left(ax+b\le0\land cx+d\lt0\right)\tiny.\]

fractional linear function has a denominator equal to #0# and #cx+d=0#. Therefore #cx+d\gt0\lor cx+d\lt0# is satisfied.

Suppose #cx+d\gt0#. According to the theorem Ordering of rational numbers #\dfrac{ax+b}{cx+d}\ge0# is true if and only if #{ax+b}\ge0#. In this case, therefore, we have #ax+b\ge0\land cx+d\gt0#.

Now assume #cx+d\lt0#. According to the last part of the above-mentioned theorem #\dfrac{ax+b}{cx+d}\ge0# holds if and only if #{ax+b}\le0#. In this case, therefore, we have #ax+b\le0\land cx+d\lt0#.

We conclude that #\left(ax+b\ge0\land cx+d\gt0\right)\lor\left(ax+b\le0\land cx+d\lt0\right)#.

Determine all #x# for which #\frac{9x-15}{5x+5}\lt -9#.

Provide your answer in the form #x \lt a\lor x\gt a# or #\left(x\lt a\right)\lor\left(x\gt b\right)# or # \left(x \gt a\right)\land \left(x \lt b\right)#, where #a\lt b#, or similar.

#x\gt -1 \land x\lt -{{5}\over{9}}#

As we cannot divide by zero, #5x+5 \ne 0#, and hence #x= -1# is not a solution to the inequality. To solve the inequality, we distinguish according to the sign of #5x+5#, that is, between #x\gt -1# and # x\lt -1#. Multilplying both sides of #\frac{9x-15}{5x+5}\lt -9# by #5x+5# in each case, we see that the given inequality is equivalent to #(x\gt -1\land 9x-15 \lt -9\cdot \left(5\cdot x+5\right)) \lor (x\lt -1\land 9x-15 \gt -9\cdot \left(5\cdot x+5\right)) #. Now the usual Rules of calculation for inequalities imply that this is equivalent to # (x\gt -1\land x \lt -{{5}\over{9}}) \lor (x\lt -1\land x \gt -{{5}\over{9}}) #. As # -1 \le -{{5}\over{9}}#, we find #x\gt -1 \land x\lt -{{5}\over{9}}#.

New example