## Why Solve Linear Equations and Inequalities?

You might be surprised to learn that applications of linear equations turn up in many places besides math classrooms. Knowing how to solve them is a basic math skill used in nearly every academic discipline and in many jobs. One of the fundamental principles of solving linear equations is that of reversing or undoing mathematical operations such as addition and subtraction. To see a linear equation in action, consider one that is used by forensic scientists to calculate blood alcohol content.

Blood alcohol content (BAC) is a measurement of how much alcohol is in someone’s blood. It is usually measured in grams and as a percentage. For example, a BAC of  $0.30\%$ is three-tenths of  $1\%$, and it indicates that there are $3$ grams of alcohol for every $1,000$ grams of blood—which is actually a lot. A BAC of  $0.05\%$ impairs reasoning and the ability to concentrate. A BAC of $0.30%$ can lead to a blackout, shortness of breath, and loss of bladder control. In most states, the legal limit for driving is a BAC of $0.08%$.

BAC is usually determined by the results of a breathalyzer, urinalysis, or blood test. Swedish physician E. M. P. Widmark developed an equation that works well for estimating BAC without using one of those tests. Widmark’s formula is widely used by forensic scientists:

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

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

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?

As you will discover, these are all questions that can be answered by solving linear equations and inequalities. Read on to learn more. At the end of the module we will revisit Joan and see how she fared.