实数的绝对值

¢ ê � ý é Š ½Â 1 é?¿¢ê x§½ÂÙýéŠ |x |Xeµ |x | =  x x > 0ž 0 x = 0ž −x x < 0ž XÃAO`²§±e�?ØÑ´3¢êS" ½n 1 |x | > 0" y² x > 0ž§|x | = x > 0¶x = 0ž§|x | = 0¶x < 0ž§|x | = −x > 0" ¤±§|x| > 0" ½n 2 x > 0ž§|x | = x" y² x > 0ž§|x | = x¶x = 0ž§|x | = 0 = x"¤±x > 0ž§|x | = x" ½n 3 x 6 0ž§|x | = −x" y² x < 0ž§|x | = −x¶x = 0ž§|x | = 0 = −x"¤± x 6 0ž§ |x | = −x" ½n 4 x 6 0ž§x = − |x |" y² Ϗ x < 0ž§|x | = −x§¤± x 6 0ž§x = − |x |" ½n 5 x 6 |x |" y² x > 0ž§x = |x |¶x < 0ž§x = − |x | < 0 < |x |"¤±§x 6 |x |" ½n 6 −x 6 |x |" y² x > 0ž§−x < 0 < x = |x |¶x 6 0ž§−x = |x |"¤±§−x 6 |x |" 2 -¢ê�ýéŠ ½n 7 x > − |x |" y² Ϗ −x 6 |x |§¤± x > − |x |" ½n 8 |x | = | −x |" y² e x > 0§K −x < 0§|x | = x = −(−x) = | −x |" e x = 0§K −x = 0§|x | = 0 = | −x |" e x < 0§K −x > 0§|x | = −x = | −x |" nþ§|x | = | −x |" ½n 9 |x− y | = | y − x |" y² |x− y | = | −(x− y) | = | y − x |" ½n 10 e x > y > 0§K |x | > | y |" y² Ϗ x > y > 0§¤± |x | = x > y = | y |" ½n 11 e x 6 y 6 0§K |x | > | y |" y² Ϗ x 6 y 6 0§¤± |x | = −x > −y = | y |" ½n 12 |x− y | = 0�¿©7‡^‡´ x = y " y² e x = y §K |x− y | = |0| = 0" ¤±§x = y ´ |x− y | = 0�¿©^‡" e |x− y | = 0§K x− y = 0§= x = y " ¤±§x = y ´ |x− y | = 0�7‡^‡" nþ§|x− y | = 0�¿©7‡^‡´ x = y " ½n 13 e a > 0§K |x | = a�¿©7‡^‡´ x = a½ x = −a" y² e x = a§K x > 0§|x | = a" e x = −a§K x 6 0§|x | = −x = −(−a) = a" ¢ê�ýéŠ - 3 ¤±§x = a½ x = −a´ |x | = a�¿©^‡" � |x | = až§ e x > 0§K x = |x | = a" e x < 0§K x = −|x| = −a" ¤±§x = a½ x = −a´ |x | = a�7‡^‡" nþ§|x | = a�¿©7‡^‡´ x = a½ x = −a" ½n 14 |x | = | y |�¿©7‡^‡´ x = y ½ x = −y " y² � x = y ½ x = −y ž§w, |x | = | y |" ¤±§x = y ½ x = −y ´ |x | = | y |�¿©^‡" Ϗ | y | > 0§¤± x = | y |½ x = − | y |" � x = | y |ž§x = y ½ x = −y " � x = − | y |ž§ x = −y ½ x = −(−y) = y " ¤±§x = y ½ x = −y ´ |x | = | y |�7‡^‡" nþ§|x | = | y |�¿©7‡^‡´ x = y ½ x = −y " ½n 15 e a > 0§K |x | 6 a�¿©7‡^‡´ −a 6 x 6 a" y² e 0 6 x 6 a§K |x | = x 6 a" e −a 6 x 6 0§K |x | = −x 6 a" ¤±§e a > 0§K −a 6 x 6 a´ |x | 6 a�¿©^‡" e x > 0§K x = |x | 6 a" e x 6 0§K x = − |x | > −a" ¤±§e a > 0§K −a 6 x 6 a´ |x | 6 a�7‡^‡" nþ§e a > 0§K |x | 6 a�¿©7‡^‡´ −a 6 x 6 a" ½n 16 e a > 0§K |x | > a�¿©7‡^‡´ x 6 −a½ x > a" 4 -¢ê�ýéŠ y² e x > a > 0§K |x | = x > a" e x 6 −a 6 0§K |x | = −x > a" ¤±§e a > 0§K x 6 −a½ x > a´ |x | > a�¿©^‡" � |x | > až§ e x > 0§K x = |x | > a" e x 6 0§K x = − |x | 6 −a" ¤±§e a > 0§K x 6 −a½ x > a´ |x | > a�7‡^‡" nþ§e a > 0§K |x | > a�¿©7‡^‡´ x 6 −a½ x > a" ½n 17 e a > 0§K |x− y | 6 a�¿©7‡^‡´ y − a 6 x 6 y + a" y² e a > 0§K |x− y | 6 a�¿©7‡^‡´ −a 6 x− y 6 a§ = y − a 6 x 6 y + a" ½n 18 e a > 0§K |x− y | > a�¿©7‡^‡´ x 6 y − a½ x > y + a" y² e a > 0§K |x− y | > a�¿©7‡^‡´ x− y 6 −a½ x− y > a§ = x 6 y − a½ x > y + a" ½n 19 |xy | = |x | · | y |" y² x > 0… y > 0ž§xy > 0§|xy | = xy = |x | · | y |" x > 0… y < 0ž§xy 6 0§|xy | = −xy = x · (−y) = |x | · | y |" x < 0… y > 0ž§xy 6 0§|xy | = −xy = (−x) · y = |x | · | y |" x < 0… y < 0ž§xy > 0§|xy | = xy = (−x) · (−y) = |x | · | y |" nþ§|xy | = |x | · | y |" ¢ê�ýéŠ - 5 ½n 20 ∣∣∣∣∣ n∏ i=1 xi ∣∣∣∣∣ = n∏ i=1 |xi |" y² n = 1ž§·Kw,¤á" b� n = k ž§·K¤á§= ∣∣∣∣∣ k∏ i=1 xi ∣∣∣∣∣ = k∏ i=1 |xi |" K n = k + 1ž§∣∣∣∣∣ k+1∏ i=1 xi ∣∣∣∣∣ = ∣∣∣∣∣ ( k∏ i=1 xi ) · xk+1 ∣∣∣∣∣ = ∣∣∣∣∣ k∏ i=1 xi ∣∣∣∣∣ · |xk+1 | = k+1∏ i=1 |xi |" ¤±·K阃��ê¤á" ½n 21 y 6= 0ž§ ∣∣∣∣ xy ∣∣∣∣ = |x || y | " y² x > 0… y > 0ž§ x y > 0§ ∣∣∣∣ xy ∣∣∣∣ = xy = |x || y | " x > 0… y < 0ž§ x y 6 0§ ∣∣∣∣ xy ∣∣∣∣ = − xy = x−y = |x || y | " x < 0… y > 0ž§ x y < 0§ ∣∣∣∣ xy ∣∣∣∣ = − xy = −xy = |x || y | " x < 0… y < 0ž§ x y > 0§ ∣∣∣∣ xy ∣∣∣∣ = xy = −x−y = |x || y | " nþ§y 6= 0ž§ ∣∣∣∣ xy ∣∣∣∣ = |x || y | " ½n 22 x 6= 0ž§∣∣x0 ∣∣ = |x |0" y² Ϗ x 6= 0ž§x0 = 1 > 0"¤±∣∣x0 ∣∣ = | 1 | = 1 = x0 = |x |0 " ½n 23 n��ê§|xn | = |x |n" y² Ϗ ∣∣∣∣∣ n∏ i=1 xi ∣∣∣∣∣ = n∏ i=1 |xi |" ¤±� x1 = x2 = x3 = · · · = xn = xž§|xn | = |x |n" 6 -¢ê�ýéŠ ½n 24 x 6= 0§n��ê§∣∣x−n ∣∣ = |x |−n" y² � x 6= 0§n��ꞧ ∣∣x−n ∣∣ = ∣∣∣∣ 1xn ∣∣∣∣ = ∣∣∣∣ ( 1x )n ∣∣∣∣ = ∣∣∣∣ 1x ∣∣∣∣n = ( 1|x | )n = 1 |x |n = |x | −n" ½n 25 x > 0§n¢ê§|xn | = |x |n" y² � x > 0§|x | = x§xn > 0"|xn | = xn = |x |n" ½n 26 |x+ y | 6 |x |+ | y |" y² Ϗ x 6 |x |§ y 6 | y |§¤± x+ y 6 |x |+ | y |" � x+ y > 0ž§|x+ y | = x+ y 6 |x |+ | y |" Ϗ −x 6 |x |§−y 6 | y |§¤± −(x+ y) 6 |x |+ | y |" � x+ y 6 0ž§|x+ y | = −(x+ y) 6 |x |+ |x |" nþ§|x+ y | 6 |x |+ | y |" ½n 27 ∣∣∣∣∣ n∑ i=1 xi ∣∣∣∣∣ 6 n∑ i=1 |xi | y² n = 1ž§·Kw,¤á" b� n = k ž§·K¤á§= ∣∣∣∣∣ k∑ i=1 xi ∣∣∣∣∣ 6 k∑ i=1 |xi |" K n = k + 1ž§∣∣∣∣∣ k+1∑ i=1 xi ∣∣∣∣∣ = ∣∣∣∣∣ ( k∑ i=1 xi ) + xk+1 ∣∣∣∣∣ 6 ∣∣∣∣∣ k∑ i=1 xi ∣∣∣∣∣+ |xk+1 | 6 k+1∑ i=1 |xi |" ¤±·K阃��ê¤á" ½n 28 |x− y | 6 |x |+ | y |" y² |x− y | = |x+ (−y) | 6 |x |+ | −y | = |x |+ | y |" ¢ê�ýéŠ - 7 ½n 29 | |x | − | y | | 6 |x+ y |" y² Ϗ |x | = | (x+ y)− y | 6 |x+ y |+ | y |§ ¤± 6 |x | − | y | 6 |x+ y |" Ϗ | y | = | (x+ y)− x | 6 |x+ y |+ |x |§ ¤± |x | − | y | > − |x+ y |" nþ§| |x | − | y | | 6 |x+ y |" ½n 30 | |x | − | y | | 6 |x− y |" y² Ϗ |x | = | (x− y) + y | 6 |x− y |+ | y |§ ¤± 6 |x | − | y | 6 |x− y |" Ϗ | y | = |x− (x− y) | 6 |x |+ |x− y |§ ¤± |x | − | y | > − |x− y |" nþ§| |x | − | y | | 6 |x− y |" ½n 31 |x− z | 6 |x− y |+ | y − z |" y² |x− z | = | (x− y) + (y − z) | 6 |x− y |+ | y − z |" ½n 32 |x− z | > | |x− y | − | y − z | |" y² |x− z | = | (x− y)− (z − y) | > | |x− y | − | z − y | | = | |x− y | − | y − z | |"