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{4x-8}{6x+2}\lt 2#.

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\lt -{{3}\over{2}} \lor x\gt -{{1}\over{3}}#

As we cannot divide by zero, #6x+2 \ne 0#, and hence #x= -{{1}\over{3}}# is not a solution to the inequality. To solve the inequality, we distinguish according to the sign of #6x+2#, that is, between #x\gt -{{1}\over{3}}# and # x\lt -{{1}\over{3}}#. Multilplying both sides of #\frac{4x-8}{6x+2}\lt 2# by #6x+2# in each case, we see that the given inequality is equivalent to #(x\gt -{{1}\over{3}}\land 4x-8 \lt 2\cdot \left(6\cdot x+2\right)) \lor (x\lt -{{1}\over{3}}\land 4x-8 \gt 2\cdot \left(6\cdot x+2\right)) #. Now the usual Rules of calculation for inequalities imply that this is equivalent to # (x\gt -{{1}\over{3}}\land x \gt -{{3}\over{2}}) \lor (x\lt -{{1}\over{3}}\land x \lt -{{3}\over{2}}) #. As # -{{1}\over{3}} \ge -{{3}\over{2}}#, we find #x\lt -{{3}\over{2}} \lor x\gt -{{1}\over{3}}#.

New example