Why it Matters: Linear Inequalities and Absolute Values

Why solve linear inequalities?

You might be surprised to learn that applications of linear inequalities 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 and inequalities is that of reversing or undoing mathematical operations such as addition and subtraction. To see a linear inequality in action, let’s consider one that’s used by forensic scientists to calculate blood alcohol content.

Not surprisingly, blood alcohol content (BAC) is a measurement of how much alcohol is in someone’s blood. It’s 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:

[latex]\text{B} = -0.015t +\left(\frac{2.84N}{Wg}\right)[/latex]

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 Anger or sadness 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	Bladder function 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’ll 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’ll revisit Joan and see how she fared.

Module 2 Learning Objectives

2.1: Linear Inequalities with One Variable

Express solutions to inequalities graphically, with interval notation, and with set-builder notation
Solve single-step linear inequalities with one variable
Combine properties of inequality to solve linear inequalities with one variable
Simplify using the Distributive Property and solve linear inequalities with one variable

2.2: Applications Using Linear Inequalities

Solve applications using linear inequalities

2.3: Describing Sets as Intersections or Unions

Find the intersection and union of two sets of numbers
Use interval notation to describe sets of numbers as intersections and unions
Recognize when an intersection has no solution or when a union has all real numbers as the solution

2.4: Solving Compound Inequalities

Solve compound inequalities of the form of OR and express the solution graphically and in interval notation (union/disjunction)
Solve compound inequalities of the form AND and express the solution graphically and in interval notation (intersection/conjunction)
Solve tripartite inequalities and express the solution graphically and in interval notation

2.5: Absolute Value Equations

Solve absolute value equations
Recognize when an absolute value equation has no solution
Solve absolute value equations containing two absolute values

2.6: Absolute Value Inequalities

Determine whether an absolute value inequality corresponds to a union or an intersection of inequalities
Solve absolute value inequalities and express the solutions graphically and in interval notation
Recognize when an absolute value inequality has no solution or all real numbers as the solution

Module 2: Linear Inequalities and Absolute Values