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.