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Mathematics LibreTexts

1.1: Why It Matters- Solving Equations and Inequalities

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    Why solve linear equations and inequalities?

    Photo of a mai tai

    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, 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:

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


    • 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
    • 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.

      Learning Outcomes

      Real Numbers

      • Classify a real number
      • Define properties of real numbers and use them to evaluate algebraic expressions

      Multi-Step Equations

      • Use properties of real numbers to solve multi-step equations
      • Define and use the distributive property to solve linear equations
      • Classify solutions to linear equations


      • Set up a linear equation from a written description of a problem and solve it
      • Use a formula to solve an application problem