SATII数学考试快递——Inequalities（2）

Now, combine the two bounds into a range of values for x. –1 ≤ x ≤ 5 is the solution.

The other solution for an absolute value inequality involves two disjoint ranges: one whose lower bound is negative infinity and whose upper bound is a real number, and one whose lower bound is a real number and whose upper bound is infinity. This occurs when the absolute value is greater than a given quantity. For example,


	Solve for x in the inequality \|3x + 4\| > 16.

First, solve for the upper range:

Then, solve for the lower range:

Now combine the two ranges to form the solution, which is two disjoint ranges: –∞ < x < –20⁄3 or 4 < x < ∞.

When working with absolute values, it is important to first isolate the expression within absolute value brackets. Then, and only then, should you solve separately for the cases in which the quantity is positive and negative.

Ranges

Inequalities are also used to express the range of values that a variable can take. a < x < b means that the value of x is greater than a and less than b. Consider the following word-problem example:


	A very complicated board game has the following recommendation on the box: “This game is only appropriate for people older than 40 but no older than 65.” What is the range of the age of people for which the board game is appropriate?

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:

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: