| x |,  x以外的符號 |  | 是所謂的絕對值. 這表示不管x是一個正數或負數的值,取絕對值得結果是恆正. 例如 | x | = 1, x = 1 x = -1,  | x | > 1, x > 1 x < -1, | x | < 1, x < 1 x > -1. 從幾何的角來看, | x | < 1可解示成以x=0為中心點距離小於1的所有點的集合. 絕對值的題目可以蠻進階的. 例如求 | 3x + 1| > | x – 4 | x的範圍? 此問題的解答, 在本文章的最下面. 首先我們來討論如何從邏輯推理的角度來探討這個問題.            | x |, the symbol , |  |, outside of x is called the absolute value.  It means that no matter x is a negative or a positive value,| x | is always positive. For example, if | x | = 1 then x = 1 or x = -1,  | x | > 1 then x > 1 or x < -1, and  | x | < 1 then x < 1 or x > -1. Geometrically, | x | < 1 means that the set of all of points that the distance from the point x = 0 is less than 1.  The problems on the absolute value could get very advanced, for example, find the range of x for | 3x + 1| > | x – 4 |.  The answer of this problem is at bottom of this article. Now, let’s find the procedure that we can solve this problem logically.

解題, 首先必先知道如何問自已一連串比此問題較簡單的問題. 就以求 | 3x + 1| > | x – 4 | x的範圍為例子. 以下一連串問題必須先瞭解. (1) | x | > 1 (2) | x + 1 | > 1 (3) | 2x + 1 | > 1 (4) | 1/x | > 1 (5) | 1 + 1/x | > 1 (6) | 1 + 1/x | > x (7) | 1 + 1/x | > | x |. 從這一連串由淺入深問題中來分析及瞭解它們邏輯的變化. 進而問自已比就以求 | 3x + 1| > | x – 4 | x的範圍更難的問題. 例如| x + 1/x | > | x || x + 1/x | +| 2x - 2/x | > | x2+x+1 |. 從以上的訓練, 可以增強邏輯推理的能力.

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