Exercise 31 Page 288

To algebraically express a verbal inequality, we will need to translate the given information into mathematical symbols and operations.

Writing the Inequality

The phrase $\text{at}$ $\text{most}$ can be expressed as $\le .$ This symbol will be at the center of our expression. On the left-hand side of the inequality symbol we will translate any verbal expression that comes before $\text{is}$ $\text{at}$ $\text{most}.$ If we let $n$ represent a number, we can form this expression. We will translate any verbal expression that comes after $\text{is}$ $\text{at}$ $\text{most}$ on the right-hand side of the inequality symbol. Finally, we can bring these two expressions together to form the inequality.

Solving the Inequality

Using the Properties of Inequality, we will solve the inequality by isolating the variable. This inequality tells us that all values less than or equal to $\text{-}8$ will satisfy the inequality.

Checking Our Solution

We can check our solution by substituting arbitrary values into the inequality translated above. The value satisfies the inequality if the inequality remains true after substituting and simplifying.

$n$	$2n+5\le n-3$	Evaluate
$\text{-}7$	$2(\text{-}7)+5\stackrel{?}{\le }\text{-}7-3$	$\text{-}9\nleqq \text{-}10$
$\text{-}8$	$2(\text{-}8)+5\stackrel{?}{\le }\text{-}8-3$	$\text{-}11\le \text{-}11$
$\text{-}9$	$2(\text{-}9)+5\stackrel{?}{\le }\text{-}9-3$	$\text{-}13\le \text{-}12$

We can conclude that as long as $n$ is less than or equal to $\text{-}8,$ the inequality is satisfied.