Let a be the age of people for which the board game is appropriate. The lower bound of a is 40, and the upper bound is 65. The range of a does not include its lower bound (it is appropriate for people “older than 40”), but it does include its upper bound (“no older than 65”, i.e., 65 is appropriate, but 66 is not). Therefore, the range of the age of people for which the board game is appropriate can be expressed by the inequality:
Here is another example:
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A company manufactures car parts. As is the case with any system of mass production, small errors occur on virtually every part. The key for this company to succeed in making viable car parts is to keep the errors within a specific range. The company knows that a particular piece they manufacture will not work if it weighs less than 98% of its target weight or more than 102% of its target weight. If the target weight of this piece is 21.5 grams, in what range of weights must the piece measure for it to function? | |
The boundary weights of this car part are .98
21.5 = 21.07 and 1.02
21.5 = 21.93 grams. The problem states that the piece cannot weigh
less than the minimum weight or
more than the maximum weight in order for it to work. This means that the part will function at boundary weights themselves, and the lower and upper bounds are included. The answer to the problem is 21.07 ≤
x ≤ 21.93, where
x is the weight of the part in grams.
Finding the range of a particular variable is essentially an exercise in close reading. Every time you come across a question involving ranges, you should carefully peruse the problem to pick out whether a particular variable’s range includes its bounds or not. This inclusion is the difference between “less than or equal to” and simply “less than.”
Operations on Ranges
Operations like addition, subtraction, and multiplication can be performed on ranges just like they can be performed on variables. For example:
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If 4 < x < 7, what is the range of 2x + 3? | |
To solve this problem, simply manipulate the range like an inequality until you have a solution. Begin with the original range:
Then multiply the inequality by 2:
Add 3 to the inequality, and you have the answer:
There is one crucial rule you need to know about multiplying ranges: if you multiply a range by a negative number, you must flip the greater-than or less-than signs. For instance, if you multiply the range 2 < x < 8 by –1, the new range will be –2 > x > –8. Math IC questions that ask you to perform operations on ranges of one variable will often test your alertness by making you multiply the range by a negative number.
Some range problems on the Math IC will be made slightly more difficult by the inclusion of more than one variable. In general, the same basic procedures for dealing with one-variable ranges applies to adding, subtracting, and multiplying two-variable ranges.
