# Identifying Inequalities as True or False

Given any two real numbers $\,x\,$ and $\,y\,,$ exactly one of the following three situations exists:

- $\,x\,$ equals $\,y\,$ (that is, $\,x\,$ and $\,y\,$ live at the same place on a real number line); or
- $\,x\,$ lies to the left of $\,y\,$ on a number line; or
- $\,x\,$ lies to the right of $\,y\,$ on a number line.

There are four mathematical sentences that make it easy to talk about the order relationships between any two real numbers: $$ \begin{gather} \cssId{s13}{x \lt y} \qquad \qquad \cssId{s14}{x \gt y}\cr\cr \cssId{s15}{x \le y} \qquad \qquad \cssId{s16}{x \ge y} \end{gather} $$

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 |

$x \lt y$ | $x\,$ is less than $\,y$ | TRUE when $\,x\,$ lies to the left of $\,y\,$ on a number line; FALSE otherwise |

$x \gt y$ | $\,x\,$ 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 $\,x \lt y\,$ or $\,x = y\,$; FALSE otherwise |

$x\ge y$ | $x\,$ is greater than or equal to $\,y$ | TRUE when $\,x \gt y\,$ or $\,x = y\,$; FALSE otherwise |

Whenever you come across a sentence of the form ‘$\,x \lt y\,$’, 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
‘$\,x \lt y\,$’
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
‘$\,x \gt y\,$’
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*.