Compound Inequalities are All Around Us.

  I am getting ready to teach compound inequalities to my Algebra I students and have been trying to think of some real world situations they might relate to.  It is fine and good to teach it using variables and such but without a context, it remains a mystery for most.

Most of the examples I've seen are ones my students will not be able to relate to such as using more gas if you drive too slow or two fast but if I change it to talk about filling the gas tank up it will make more sense.  If the tank takes a maximum of 5 gallons gas in their ATV, then they will add between 0 and 5 gallons or 0 <x< 5. The above example will make more sense to them because its a better context.

Another example is one involved video games.  On the level they have a certain number of tries to eliminate all the trees in front of them.  Say they are given 6 bombs to accomplish the feat so the compound inequality would be 0 <x< 6 because once they run out, they cannot do more.  This type of inequality appears in numerous video games.

You are planning a trip.  You are looking at going to either Mexico or Hawaii and the airfare is about the same but the hotel room cost is quite different.  The room in Mexico is $75 per night while the room in Hawaii is $152 per night so you have an or.  x > $75 or x> $152 and the x is based on how many nights you will spend there based on a minimum of one night. This is an example from my life.  I ended up going to Mexico due to the cheaper cost of hotel rooms for the Christmas Holidays.

Another example is when looking at renting cars.  Depending on the time, the company, and length, one rental company may be a much better deal.  For instance, Hertz may offer you a full sized car for $87.50 per day while Avis has the same for for $92.00 so your inequality becomes x > $87.50 or x > $92.00.

You always see examples that are like x > 32 or x < -15 so what would that be in a real world situation?  If you think of the minus sign as a direction, then you can look at it this way.  You are visiting your aunt in Fresno.  You want to go to a game store but you are between two.  One is 32 minutes heading towards the mountains, or 15 minutes towards the ocean.  These do not involve equals because anyone who has driven in traffic knows the time is usually the minimum time. Another scenario for x > 32 or x - 15 is that my friend and I went on a diet, she lost at least 15 pounds on it while I gained 32 pounds using the same diet.

One compound inequality I see all the time involves buying things on sale at fabric or grocery stores. When fabric stores offer patterns on sale for $1.99, they put a limit of 10 patterns on it or 0 <x < 10.  Often there are limits on butter at $1.69 per pound instead of $2.69 per pound.  You might be restricted to 5 pounds.

Even when traveling on the airline, you are allowed to have up to 50 pounds in your luggage so it has to be between 0 and 50 or they might charge you an additional fee.  Most airlines also allow you so many suitcases free before you are charged for extra bags.  Some airlines in Alaska do not charge by the piece, they charge by a total weight for your check through and carry on bags.  I've used an airline that had a maximum of 100 pounds before they charge you almost $1.00 per pound overage.  So you made sure the weight was no more than 100 pounds.

So as you can see, the world is filled with lots of compound inequalities.  Let me know what you think, I'd love to hear.  Have a great day.