Putting it Together: Linear Inequalities
At the start of this module, we continued the story of Joan who had been drinking at a party. She was ready to drive home when we posed a question:
- Given any amount that she drinks, can you figure out how much time should pass before she can drive safely and legally?
Previously, we used the same formula commonly used by forensic scientists to calculate blood alcohol content to find her blood alcohol content to determine whether she was within the legal limit to drive:
[latex]B=-0.015t +\left(\dfrac{2.84N}{Wg}\normalsize\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 [latex]12[/latex]-ounce beer, one [latex]5[/latex]-ounce glass of wine, or one [latex]1.5[/latex]-ounce shot of liquor). N should be at least [latex]1[/latex].
- W = weight in pounds
- g = gender constant: [latex]0.68[/latex] for men and [latex]0.55[/latex] for women
- Given any amount that she drinks, can you figure out how much time should pass before she can drive safely and legally?
[latex]\text{0.08% }\lt -0.015t +\left(\dfrac{2.84N}{Wg}\normalsize\right)[/latex]
or
[latex]\text{0.08% }\gt -0.015t +\left(\dfrac{2.84N}{Wg}\normalsize\right)[/latex]
Using words to translate what these expressions mean is often helpful:
[latex]\text{0.08% }\lt -0.015t +\left(\dfrac{2.84N}{Wg}\normalsize\right)[/latex]
Translation: For a limit of [latex]0.08\%[/latex], what time will make this side greater than [latex]0.08\%[/latex]?
That doesn't seem right. We want to know how much time will make the right-hand side less than [latex]0.08\%[/latex], so we would want to choose the other one:
[latex]\text{0.08% }\gt -0.015t +\left(\dfrac{2.84N}{Wg}\normalsize\right)[/latex]
Solving for time gives us the following:
[latex]\begin{array}{c}\text{0.08 }\gt -0.015t +0.1147\\{-0.0347}\gt -0.015t\end{array}[/latex]
Divide both sides by [latex]-0.015[/latex] and remember to change the direction of the inequality sign. That gives us the following:
[latex]2.31<t[/latex]
We have our answer to this new question. According to the expression above, our answer is "any time greater than [latex]2.3[/latex] hours will be long enough for Joan to be within the legal limit for driving after she's consumed three standard drinks." We hope Joan takes our advice and gets home okay.
Progressive Effects of Alcohol | ||
---|---|---|
BAC (% by vol.) | Behavior | Impairment |
[latex]0.001–0.029[/latex] |
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[latex]0.030–0.059[/latex] |
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[latex]0.060–0.099[/latex] |
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[latex]0.100–0.199[/latex] |
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[latex]0.200–0.299[/latex] |
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[latex]0.300–0.399[/latex] |
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[latex]0.400–0.500[/latex] |
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[latex]>0.50[/latex] |
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