How do you solve #(3-x)/(x+5)<=0# using a sign chart?

1 Answer
Oct 2, 2016

Answer:

#(3-x)/(x+5) <= 0# when either #x < -5# or #x >= 3#, which is the solution for the inequality. Note that at #x=-5#, it is not defined.

Explanation:

The sign of #(3-x)/(x+5)# depends on the signs of binomials #(3-x)# and #x+5#, which should change around the values #-5# and #3# respectively. In sign chart we divide the real number line using these values, i.e. below #-5#, between #-5# and #3# and above #3# and see how the sign of #(3-x)/(x+5)# changes.

Sign Chart

#color(white)(XXXXXXXXXXX)-5color(white)(XXXXX)3#

#(3-x)color(white)(XXX)+ive color(white)(XXXX)+ive color(white)(XXXX)-ive#

#(x+5)color(white)(XXX)-ive color(white)(XXXX)+ive color(white)(XXXX)+ive#

#(3-x)/(x+5)color(white)(XXX)-ive color(white)(XXXX)+ive color(white)(XXXX)-ive#

It is observed that #(3-x)/(x+5) <= 0# when either #x < -5# or #x >= 3#, which is the solution for the inequality. Note that at #x=-5#, it is not defined.
graph{(3-x)/(x+5) [-20, 20, -10, 10]}