Inequalities(3)

　　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 美国GREatest is the upper bound. Therefore, –12 < jk < 48.　　Let’s try one more example of performing operations on ranges:

　　If 3 ≤ x < 7 and , 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.

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