Absolute Value Inequalities 06:15 minutes

Video Transcript

Transcript Absolute Value Inequalities

Daisy plans to go to the Super Bowl parade, so she wants to skip school. Since her parents will definitely not be fond of her plans, she’ll pretend to be sick with a fever and sneak out of the house after her parents leave for work. In biology class, Daisy learned that the average human body temperature is 98.6° Fahrenheit, but can vary by ±1°. How can she calculate a temperature that will make her appear to be too sick to go to school?

Absolute Value Inequalities

To figure this out, Daisy decides to use Absolute Value Inequalities. So the average body temperature is 98.6° Fahrenheit and can vary by ±1°. Let’s write this as an absolute value inequality. We’ll use 'x' to represent the highs and lows of a healthy temperature range. The |x - 98.6| ≤ 1. When you have an equation containing the less than sign and the absolute value is on the left side of the equation, the solution should be written with two statements that can be joined by the word ‘and’. Let's see what that would look like: (the negative value) ≤ x ≤ (the positive value). Notice how the absolute value bars are removed.
Now to solve Daisy's problem. Using the same format, we can write -1 ≤ x -98.6 ≤ 1. It's easier to calculate this problem if we write it as two inequalities: x - 98.6 ≥ -1 AND x - 98.6 ≤ 1. Now solve by isolating the variable. Let's add 98.6 to both sides of each inequality. There are two expressions: x - 98.6 ≥ -1 and x - 98.6 ≤ 1, giving us x ≥ 97.6 and x ≤ 99.6, respectively.
So, in plain English, this means that a temperature as low as 97.6 and as high as 99.6 wouldn't be enough to keep Daisy home sick. You can write the solution to this absolute value inequality in several ways. You can write it as two inequalities joined by the word 'and,' or like this using the less than operator. The solution set is all the numbers between 97.6 and 99.6, inclusive. Inclusive is equivalent to drawing a closed circle on a number line. Notice how the inequality is displayed on the thermometer. The healthy zone is shown on one section of the thermometer.

Inequalities with a greater than sign

Daisy also knows that when the body temperature varies by greater than or equal to six degrees Fahrenheit, it’s considered to be a medical emergency. She doesn’t want to end up in the hospital, so she needs to know the dangerous temperature range. Let's write the inequality for the danger zone.
The |x - 98.6| ≥ 6. When you have an equation containing the greater than sign and the absolute value is on the left side of the equation, the solution should be written with two statements that can be joined by the word OR. Like this. And the format is 'x' is less than or equal to the negative value or ‘x' is greater than or equal to the positive value. Notice the absolute value bars are gone.
Ok let's help Daisy figure out how to solve this problem. Use the format to write two inequalities. x - 98.6 ≤ -6 or x - 98.6 ≥ 6 then solve by isolating the variable. The solution for the |x - 98.6| ≥ 6 is: x ≤ 92.6 or x ≥ 104.6. Notice how the solution set is displayed on the thermometer. You have closed circles at 92.6 and 104.6. With an or situation, the solution set is always divided into two pieces.

Okay, lets put all this information together so Daisy can figure out what temperature will make her appear too sick to go to school. The healthy zone is greater than or equal to 97.6 and less than or equal to 99.6. The danger zone is less than or equal to 92.6 OR greater than or equal to 104.6. Remember, if the |x| < or ≤ a, you use the compound inequality with ‘and’, if the absolute value of x is > or ≥ a, you use the compound inequality with ‘or’.

Luckily, Daisy knows just the right place to get the perfect thermometer reading...

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    Posted by Michel H., about 1 year ago