SATII数学考试快递——Inequalities(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.
Addition with Ranges of Two or More Variables
|
Simply add the ranges. The lower bound is –2 + 0 = –2. The upper bound is 8 + 5 = 13. Therefore, –2 < x + y < 13.
Subtraction with Ranges of Two or More Variables
|
In this case, you have to find the range of –t. By multiplying the range of t by –1 and reversing the direction of the inequalities, we find that 1 < –t < 3. Now we can simply add the ranges again to find the range of s – t. 4 + 1 = 5, and 7 + 3 = 10. Therefore, 5 < s – t < 10.
In general, to subtract ranges, find the range of the opposite of the variable being subtracted, and then add the ranges as usual.
Multiplication with Ranges of Two or More Variables
|
First, multiply the lower bound of one variable by the lower and upper bounds of the other variable:
Then, multiply the upper bound of one variable with both bounds of the other variable:
The least of these four products becomes the lower bound, and the greatest is the upper bound. Therefore, –12 < jk < 48.
Let’s try one more example of performing operations on ranges:
|
, what is the range of 2(x + y)?The first step is to find the range of x + y. Notice that the range of y is written backward, with the upper bound to the left of the variable. Rewrite it first:
Next add the ranges to find the range of x + y:
We have our bounds for the range of x + y, but are they included in the range? In other words, is the range 0 < x + y < 11, 0 ≤ x + y ≤ 11, or some combination of these two?
The rule to answer this question is the following: if either of the bounds that are being added, subtracted, or multiplied is non-inclusive (< or >), then the resulting bound is non-inclusive. Only when both bounds being added, subtracted, or multiplied are inclusive (≤ or ≥) is the resulting bound also inclusive.
The range of x includes its lower bound, 3, but not its upper bound, 7. The range of y includes both its bounds. Therefore, the range of x + y is 0 ≤ x + y < 11, and the range of 2(x + y) is 0 ≤ 2(x + y) < 22.