## Why Learn about Linear Inequalities?

Previously, we introduced Widmark’s formula for calculating blood alcohol content:

$\text{B} = -0.015t +\left(\dfrac{2.84N}{Wg}\normalsize\right)$

where

• B = percentage of BAC
• t = number of hours since the first drink
• N = number of “standard drinks” (a standard drink is one $12$-ounce beer, one $5$-ounce glass of wine, or one $1.5$-ounce shot of liquor). N should be at least $1$.
• W = weight in pounds
• g = gender constant: $0.68$ for men and $0.55$ for women

Remember Joan? When we left her, she was at a party. She had three drinks and then wanted to leave. This was worrisome. We posed the following questions:

1. Where would she fall on the table of the progressive effects of alcohol after $1.5$ hours?
2. Would she be within the legal limit to drive after this amount of time?

We found that Joan’s BAC was $0.092\%$ and by using the table below, she’s probably experiencing blunted feelings, reduced sensitivity to pain, euphoria, disinhibition, and extroversion. In addition, her reasoning, depth perception, peripheral vision, and glare recovery are probably impaired.

Additionally, we determined that she would not be within the legal limit to drive assuming the legal limit is $0.08\%$.  Joan needs a ride home, or she needs to wait until she has metabolized the alcohol in her system so she can safely drive.

Now we will pose another question that may help her plan how long she needs to wait before she drives after drinking at a party.

1. Given any amount that she drinks, can you figure out how much time should pass before she can drive safely and legally?

We solved for BAC given three standard drinks, $1.5$ hours, and Joan’s weight. What is different about this question? What are we solving, an equation or an inequality?

After working through the readings, videos and problems sets in this module, you will hopefully realize that you can answer this question with an inequality. We will revisit the answer to this question at the end of the module.

In the following table, the progressive effects of alcohol are defined for ranges of blood alcohol content.

Progressive effects of alcohol
BAC (% by vol.) Behavior Impairment
$0.001–0.029$
• Average individual appears normal
• Subtle effects that can be detected with special tests
$0.030–0.059$
• Mild euphoria
• Relaxation
• Joyousness
• Talkativeness
• Decreased inhibition
• Concentration
$0.060–0.099$
• Blunted feelings
• Reduced sensitivity to pain
• Euphoria
• Disinhibition
• Extroversion
• Reasoning
• Depth perception
• Peripheral vision
• Glare recovery
$0.100–0.199$
• Overexpression
• Boisterousness
• Possibility of nausea and vomiting
• Reflexes
• Reaction time
• Gross motor control
• Staggering
• Slurred speech
• Temporary erectile dysfunction
$0.200–0.299$
• Nausea
• Vomiting
• Emotional swings
• Partial loss of understanding
• Impaired sensations
• Decreased libido
• Possibility of stupor
• Severe motor impairment
• Loss of consciousness
• Memory blackout
$0.300–0.399$
• Stupor
• Central nervous system depression
• Loss of understanding
• Lapses in and out of consciousness
• Low possibility of death
• Breathing
• Dysequilibrium
• Heart rate
$0.400–0.500$
• Severe central nervous system depression
• Coma
• Possibility of death
• Breathing
• Heart rate
• Positional Alcohol Nystagmus
$>0.50$
• High risk of poisoning
• High possibility of death
• Life

Joan likes to party and believes she is “just fine” when it comes to driving. At a party, though, she downs three standard drinks, one after the other, and then decides to leave. If Joan weighs $135$ pounds, where would she be on the table of the progressive effects of alcohol after $1.5$ hours? Would she be within the legal limit to drive home after this amount of time? Given any amount that she drinks, can you figure out how long she must wait before she can drive safely and legally?