Video Transcript
Find algebraically the solution set
of the inequality the absolute value of six minus 𝑥 is less than three.
We begin by recalling that the
absolute value of a number is its distance from zero on a number line. In this question, we are told that
the absolute value of six minus 𝑥 is less than three, which means that six minus 𝑥
lies between negative three and three. Note that the question gives us a
strict inequality, so negative three and three are not included. We can then express this as a
double inequality: six minus 𝑥 is greater than negative three and less than
three. Subtracting six from each part of
the inequality, we have negative 𝑥 is greater than negative nine and less than
negative three. We can then divide through by
negative one, recalling that when multiplying or dividing by a negative number, this
changes the direction of the inequality symbol. For example, a less than symbol
becomes a greater than symbol.
We can therefore conclude that if
the absolute value of six minus 𝑥 is less than three, then 𝑥 is greater than three
and less than nine. As we are asked to give our answer
using set notation, we have the open interval from three to nine. In order to check our solution, it
is worth substituting a value from in our interval into the original inequality. If we let 𝑥 equals four, the
left-hand side becomes the absolute value of six minus four. This is equal to the absolute value
of two, which in turn is equal to two. And as this value is less than
three, this suggests that our solution set of the open interval from three to nine
is correct.
