Given any two real numbers
$\,x\,$ and $\,y\,$,
exactly one of
the following three situations exists:
There are four mathematical sentences that make it easy to talk about the order relationships between any two real numbers: $$ \cssId{s13}{x < y} \qquad \qquad \qquad \cssId{s14}{x > y} \qquad \qquad \qquad \cssId{s15}{x \le y} \qquad \qquad \qquad \cssId{s16}{x \ge y} $$
As with all mathematical sentences,
you should know
the correct way to read each of these sentences,
and
the condition(s) under which each is true or false.
This information is summarized below.
sentence | how to read | truth of sentence |
$x < y$ | $x\,$ is less than $\,y$ | TRUE when $\,x\,$ lies to the left of $\,y\,$ on a number line; FALSE otherwise |
$x > y$ | $\,x\,$ is greater than $\,y\,$ | TRUE when $\,x\,$ lies to the right of $\,y\,$ on a number line; FALSE otherwise |
$x\le y$ | $x\,$ is less than or equal to $\,y$ | TRUE when $\,x < y\,$ or $\,x = y\,$; FALSE otherwise |
$x\ge y$ | $x\,$ is greater than or equal to $\,y$ | TRUE when $\,x > y\,$ or $\,x = y\,$; FALSE otherwise |
Whenever you come across a sentence of the
form ‘$\,x < y\,$’, think to yourself:
does
$\,x\,$ lie to the left of $\,y\,$ on a number line?
Whenever you come across a sentence of the
form ‘$\,x > y\,$’, think to yourself:
does
$\,x\,$ lie to the right of $\,y\,$ on a number line?
CAUTION!
DO NOT read the sentence ‘$\,x < y\,$’ as
‘$\,x\,$ is smaller than $\,y\,$’.
Being “smaller than” and being “less than”
are two different ideas:
smaller than means closer to zero;
less than means farther to the left on a number line.
Similarly, DO NOT read the sentence ‘$\,x > y\,$’ as
‘$\,x\,$ is bigger than $\,y\,$’.
Being “bigger than” and being “greater than” are two different ideas:
bigger than means farther away from zero;
greater than means farther to the right on a number line.