$2$

ORDER FOR THE REAL NUMBERS $($continued$)$

EXAMPLES Write a different inequality with the same meaning.

The inequality $-5>$a has the same meaning

$9.-3>-8$ The inequality has the same meaning.

Expressions like $ab$ and $ab$ are also inequalities. We read $ab$ as $"a$ is less than or equal to $b."$ We read $ab$ as $"a$ is greater than or equal to $b."$ If $a$ is nonnegative, then $a0.$

EXAMPLES Determine whether each of the following is true or false.

$10.-85\mathrm{.}7$ True since $-8<mn\; id="EditorElement\_651473">5</mn><mn\; id="EditorElement\_651474">.</mn><mn\; id="EditorElement\_651475">7</mn>$ is true.

$11.-8-8$ True since $-8=-8$ is true.

$12.-74\frac{1}{3},$ False since neither $-7>4\frac{1}{3}$ nor $-7=4\frac{1}{3}$ is true.

$13.-\frac{2}{3}-\frac{5}{4},$ True since $-\frac{2}{3}=-0\mathrm{.}666$dots and $-\frac{5}{4}=-1\mathrm{.}25$ and $-0\mathrm{.}666$dots$>-1\mathrm{.}25.$

Do Exercises $1-16.$

$($Exercises $1-10$ are on the preceding page.$)$

$3$

GRAPHING INEQUALITIES ON THE NUMBER LINE

A replacement that makes an inequality true is called a solution. The set of all solutions is called the solution set. A graph of an inequality is a drawing that represents its solution set.

EXAMPLE $1$ Graph the inequality $x>-3$ on the number line.

The solutions consist of all real numbers greater than $-3,$ so we shade all numbers greater than $-3.$ Since $-3$ is not a solution, we use a parenthesis at $-3$ to indicate this. The graph represents the solution set $\{x|x>-3\}.$

EXAMPLE $2$ Graph the inequality $x2$ on the number line.

We make a drawing that represents the solution set $\{x|x2\}.$ The graph consists of $2$ as well as the numbers less than $2.$ We shade all numbers to the left of $2$ and use a bracket at $2$ to indicate that it is also a solution.

EXERCISES

Write a different inequality with the same meaning.

Determine whether each $of$ the following $is$ true $or$ false.

$3.6-9\mathrm{.}4$

$4.-18-18$

$15.-7\mathrm{.}6-10\frac{4}{5}$

$16.-\frac{24}{27}-\frac{25}{28}$