id	sid	tid	token	lemma	pos
ajst-9878	1	1	academic	academic	ADJ
ajst-9878	1	2	journal	journal	NOUN
ajst-9878	1	3	of	of	ADP
ajst-9878	1	4	science	science	NOUN
ajst-9878	1	5	and	and	CCONJ
ajst-9878	1	6	technology	technology	NOUN
ajst-9878	1	7	issn	issn	NOUN
ajst-9878	1	8	:	:	PUNCT
ajst-9878	1	9	2771	2771	NUM
ajst-9878	1	10	-	-	SYM
ajst-9878	1	11	3032	3032	NUM
ajst-9878	1	12	|	|	NOUN
ajst-9878	1	13	vol	vol	NOUN
ajst-9878	1	14	.	.	PROPN
ajst-9878	2	1	6	6	NUM
ajst-9878	2	2	,	,	PUNCT
ajst-9878	2	3	no	no	INTJ
ajst-9878	2	4	.	.	NOUN
ajst-9878	2	5	2	2	NUM
ajst-9878	2	6	,	,	PUNCT
ajst-9878	2	7	2023	2023	NUM
ajst-9878	2	8	164	164	NUM
ajst-9878	2	9	study	study	NOUN
ajst-9878	2	10	of	of	ADP
ajst-9878	2	11	the	the	DET
ajst-9878	2	12	application	application	NOUN
ajst-9878	2	13	of	of	ADP
ajst-9878	2	14	inequalities	inequality	NOUN
ajst-9878	2	15	in	in	ADP
ajst-9878	2	16	mathematics	mathematics	PROPN
ajst-9878	2	17	mingzhe	mingzhe	PROPN
ajst-9878	2	18	sun1	sun1	PROPN
ajst-9878	2	19	,	,	PUNCT
ajst-9878	2	20	a	a	PRON
ajst-9878	2	21	,	,	PUNCT
ajst-9878	2	22	heng	heng	PROPN
ajst-9878	2	23	cao1	cao1	PROPN
ajst-9878	2	24	1weifang	1weifang	NUM
ajst-9878	2	25	no.1	no.1	PROPN
ajst-9878	2	26	middle	middle	ADJ
ajst-9878	2	27	school	school	NOUN
ajst-9878	2	28	,	,	PUNCT
ajst-9878	2	29	weifang	weifang	PROPN
ajst-9878	2	30	261205	261205	NUM
ajst-9878	2	31	,	,	PUNCT
ajst-9878	2	32	china	china	PROPN
ajst-9878	2	33	a845606557@qq.com	a845606557@qq.com	PROPN
ajst-9878	2	34	abstract	abstract	PROPN
ajst-9878	2	35	:	:	PUNCT
ajst-9878	2	36	inequalities	inequality	NOUN
ajst-9878	2	37	have	have	VERB
ajst-9878	2	38	a	a	DET
ajst-9878	2	39	very	very	ADV
ajst-9878	2	40	important	important	ADJ
ajst-9878	2	41	place	place	NOUN
ajst-9878	2	42	in	in	ADP
ajst-9878	2	43	the	the	DET
ajst-9878	2	44	study	study	NOUN
ajst-9878	2	45	of	of	ADP
ajst-9878	2	46	mathematics	mathematic	NOUN
ajst-9878	2	47	,	,	PUNCT
ajst-9878	2	48	and	and	CCONJ
ajst-9878	2	49	their	their	PRON
ajst-9878	2	50	use	use	NOUN
ajst-9878	2	51	can	can	AUX
ajst-9878	2	52	be	be	AUX
ajst-9878	2	53	very	very	ADV
ajst-9878	2	54	convenient	convenient	ADJ
ajst-9878	2	55	in	in	ADP
ajst-9878	2	56	solving	solve	VERB
ajst-9878	2	57	many	many	ADJ
ajst-9878	2	58	practical	practical	ADJ
ajst-9878	2	59	problems	problem	NOUN
ajst-9878	2	60	in	in	ADP
ajst-9878	2	61	our	our	PRON
ajst-9878	2	62	daily	daily	ADJ
ajst-9878	2	63	lives	life	NOUN
ajst-9878	2	64	.	.	PUNCT
ajst-9878	3	1	inequalities	inequality	NOUN
ajst-9878	3	2	have	have	AUX
ajst-9878	3	3	become	become	VERB
ajst-9878	3	4	an	an	DET
ajst-9878	3	5	important	important	ADJ
ajst-9878	3	6	part	part	NOUN
ajst-9878	3	7	of	of	ADP
ajst-9878	3	8	the	the	DET
ajst-9878	3	9	basic	basic	ADJ
ajst-9878	3	10	theory	theory	NOUN
ajst-9878	3	11	of	of	ADP
ajst-9878	3	12	mathematics	mathematic	NOUN
ajst-9878	3	13	and	and	CCONJ
ajst-9878	3	14	have	have	AUX
ajst-9878	3	15	attracted	attract	VERB
ajst-9878	3	16	many	many	ADJ
ajst-9878	3	17	professionals	professional	NOUN
ajst-9878	3	18	to	to	PART
ajst-9878	3	19	study	study	VERB
ajst-9878	3	20	them	they	PRON
ajst-9878	3	21	.	.	PUNCT
ajst-9878	4	1	the	the	DET
ajst-9878	4	2	use	use	NOUN
ajst-9878	4	3	of	of	ADP
ajst-9878	4	4	inequalities	inequality	NOUN
ajst-9878	4	5	provides	provide	VERB
ajst-9878	4	6	an	an	DET
ajst-9878	4	7	intuitive	intuitive	ADJ
ajst-9878	4	8	mathematical	mathematical	ADJ
ajst-9878	4	9	model	model	NOUN
ajst-9878	4	10	for	for	ADP
ajst-9878	4	11	understanding	understand	VERB
ajst-9878	4	12	inequalities	inequality	NOUN
ajst-9878	4	13	in	in	ADP
ajst-9878	4	14	the	the	DET
ajst-9878	4	15	real	real	ADJ
ajst-9878	4	16	world	world	NOUN
ajst-9878	4	17	,	,	PUNCT
ajst-9878	4	18	making	make	VERB
ajst-9878	4	19	it	it	PRON
ajst-9878	4	20	easier	easy	ADJ
ajst-9878	4	21	for	for	SCONJ
ajst-9878	4	22	us	we	PRON
ajst-9878	4	23	to	to	PART
ajst-9878	4	24	learn	learn	VERB
ajst-9878	4	25	mathematics	mathematic	NOUN
ajst-9878	4	26	and	and	CCONJ
ajst-9878	4	27	solve	solve	VERB
ajst-9878	4	28	many	many	ADJ
ajst-9878	4	29	practical	practical	ADJ
ajst-9878	4	30	problems	problem	NOUN
ajst-9878	4	31	in	in	ADP
ajst-9878	4	32	our	our	PRON
ajst-9878	4	33	lives	life	NOUN
ajst-9878	4	34	.	.	PUNCT
ajst-9878	5	1	in	in	ADP
ajst-9878	5	2	this	this	DET
ajst-9878	5	3	paper	paper	NOUN
ajst-9878	5	4	,	,	PUNCT
ajst-9878	5	5	we	we	PRON
ajst-9878	5	6	explore	explore	VERB
ajst-9878	5	7	the	the	DET
ajst-9878	5	8	use	use	NOUN
ajst-9878	5	9	of	of	ADP
ajst-9878	5	10	mathematical	mathematical	ADJ
ajst-9878	5	11	inequalities	inequality	NOUN
ajst-9878	5	12	in	in	ADP
ajst-9878	5	13	everyday	everyday	ADJ
ajst-9878	5	14	learning	learning	NOUN
ajst-9878	5	15	.	.	PUNCT
ajst-9878	6	1	keywords	keyword	NOUN
ajst-9878	6	2	:	:	PUNCT
ajst-9878	6	3	inequalities	inequality	NOUN
ajst-9878	6	4	,	,	PUNCT
ajst-9878	6	5	learning	learn	VERB
ajst-9878	6	6	strategies	strategy	NOUN
ajst-9878	6	7	,	,	PUNCT
ajst-9878	6	8	applied	apply	VERB
ajst-9878	6	9	research	research	NOUN
ajst-9878	6	10	.	.	PUNCT
ajst-9878	7	1	1	1	X
ajst-9878	7	2	.	.	X
ajst-9878	7	3	introduction	introduction	NOUN
ajst-9878	7	4	inequalities	inequality	NOUN
ajst-9878	7	5	can	can	AUX
ajst-9878	7	6	be	be	AUX
ajst-9878	7	7	used	use	VERB
ajst-9878	7	8	to	to	PART
ajst-9878	7	9	solve	solve	VERB
ajst-9878	7	10	problems	problem	NOUN
ajst-9878	7	11	in	in	ADP
ajst-9878	7	12	mathematics	mathematic	NOUN
ajst-9878	7	13	such	such	ADJ
ajst-9878	7	14	as	as	ADP
ajst-9878	7	15	finding	find	VERB
ajst-9878	7	16	the	the	DET
ajst-9878	7	17	most	most	ADJ
ajst-9878	7	18	value	value	NOUN
ajst-9878	7	19	,	,	PUNCT
ajst-9878	7	20	comparing	compare	VERB
ajst-9878	7	21	sizes	size	NOUN
ajst-9878	7	22	and	and	CCONJ
ajst-9878	7	23	finding	find	VERB
ajst-9878	7	24	a	a	DET
ajst-9878	7	25	range	range	NOUN
ajst-9878	7	26	of	of	ADP
ajst-9878	7	27	values	value	NOUN
ajst-9878	7	28	to	to	PART
ajst-9878	7	29	prove	prove	VERB
ajst-9878	7	30	a	a	DET
ajst-9878	7	31	conclusion	conclusion	NOUN
ajst-9878	7	32	.	.	PUNCT
ajst-9878	8	1	common	common	ADJ
ajst-9878	8	2	ideas	idea	NOUN
ajst-9878	8	3	include	include	VERB
ajst-9878	8	4	:	:	PUNCT
ajst-9878	8	5	using	use	VERB
ajst-9878	8	6	basic	basic	ADJ
ajst-9878	8	7	inequalities	inequality	NOUN
ajst-9878	8	8	,	,	PUNCT
ajst-9878	8	9	using	use	VERB
ajst-9878	8	10	function	function	NOUN
ajst-9878	8	11	monotonicity	monotonicity	NOUN
ajst-9878	8	12	,	,	PUNCT
ajst-9878	8	13	using	use	VERB
ajst-9878	8	14	deflation	deflation	NOUN
ajst-9878	8	15	,	,	PUNCT
ajst-9878	8	16	etc	etc	X
ajst-9878	8	17	.	.	X
ajst-9878	9	1	in	in	ADP
ajst-9878	9	2	addition	addition	NOUN
ajst-9878	9	3	to	to	ADP
ajst-9878	9	4	explaining	explain	VERB
ajst-9878	9	5	the	the	DET
ajst-9878	9	6	theory	theory	NOUN
ajst-9878	9	7	of	of	ADP
ajst-9878	9	8	inequalities	inequality	NOUN
ajst-9878	9	9	to	to	ADP
ajst-9878	9	10	students	student	NOUN
ajst-9878	9	11	and	and	CCONJ
ajst-9878	9	12	raising	raise	VERB
ajst-9878	9	13	their	their	PRON
ajst-9878	9	14	awareness	awareness	NOUN
ajst-9878	9	15	of	of	ADP
ajst-9878	9	16	the	the	DET
ajst-9878	9	17	application	application	NOUN
ajst-9878	9	18	of	of	ADP
ajst-9878	9	19	inequalities	inequality	NOUN
ajst-9878	9	20	to	to	PART
ajst-9878	9	21	solve	solve	VERB
ajst-9878	9	22	problems	problem	NOUN
ajst-9878	9	23	,	,	PUNCT
ajst-9878	9	24	you	you	PRON
ajst-9878	9	25	should	should	AUX
ajst-9878	9	26	also	also	ADV
ajst-9878	9	27	do	do	VERB
ajst-9878	9	28	a	a	DET
ajst-9878	9	29	good	good	ADJ
ajst-9878	9	30	job	job	NOUN
ajst-9878	9	31	of	of	ADP
ajst-9878	9	32	demonstrating	demonstrate	VERB
ajst-9878	9	33	the	the	DET
ajst-9878	9	34	application	application	NOUN
ajst-9878	9	35	of	of	ADP
ajst-9878	9	36	inequalities	inequality	NOUN
ajst-9878	9	37	so	so	SCONJ
ajst-9878	9	38	that	that	SCONJ
ajst-9878	9	39	they	they	PRON
ajst-9878	9	40	can	can	AUX
ajst-9878	9	41	accumulate	accumulate	VERB
ajst-9878	9	42	relevant	relevant	ADJ
ajst-9878	9	43	application	application	NOUN
ajst-9878	9	44	experience	experience	NOUN
ajst-9878	9	45	.	.	PUNCT
ajst-9878	10	1	the	the	DET
ajst-9878	10	2	students	student	NOUN
ajst-9878	10	3	should	should	AUX
ajst-9878	10	4	also	also	ADV
ajst-9878	10	5	be	be	AUX
ajst-9878	10	6	able	able	ADJ
ajst-9878	10	7	to	to	PART
ajst-9878	10	8	gain	gain	VERB
ajst-9878	10	9	experience	experience	NOUN
ajst-9878	10	10	in	in	ADP
ajst-9878	10	11	the	the	DET
ajst-9878	10	12	application	application	NOUN
ajst-9878	10	13	of	of	ADP
ajst-9878	10	14	inequalities	inequality	NOUN
ajst-9878	10	15	.	.	PUNCT
ajst-9878	11	1	the	the	DET
ajst-9878	11	2	use	use	NOUN
ajst-9878	11	3	of	of	ADP
ajst-9878	11	4	mathematical	mathematical	ADJ
ajst-9878	11	5	ideas	idea	NOUN
ajst-9878	11	6	to	to	PART
ajst-9878	11	7	solve	solve	VERB
ajst-9878	11	8	problems	problem	NOUN
ajst-9878	11	9	with	with	ADP
ajst-9878	11	10	inequalities	inequality	NOUN
ajst-9878	11	11	begins	begin	VERB
ajst-9878	11	12	with	with	ADP
ajst-9878	11	13	the	the	DET
ajst-9878	11	14	idea	idea	NOUN
ajst-9878	11	15	of	of	ADP
ajst-9878	11	16	categorical	categorical	ADJ
ajst-9878	11	17	discussion	discussion	NOUN
ajst-9878	11	18	,	,	PUNCT
ajst-9878	11	19	which	which	PRON
ajst-9878	11	20	allows	allow	VERB
ajst-9878	11	21	complex	complex	ADJ
ajst-9878	11	22	mathematical	mathematical	ADJ
ajst-9878	11	23	knowledge	knowledge	NOUN
ajst-9878	11	24	to	to	PART
ajst-9878	11	25	be	be	AUX
ajst-9878	11	26	divided	divide	VERB
ajst-9878	11	27	into	into	ADP
ajst-9878	11	28	smaller	small	ADJ
ajst-9878	11	29	problems	problem	NOUN
ajst-9878	11	30	,	,	PUNCT
ajst-9878	11	31	so	so	SCONJ
ajst-9878	11	32	that	that	SCONJ
ajst-9878	11	33	by	by	ADP
ajst-9878	11	34	solving	solve	VERB
ajst-9878	11	35	partial	partial	ADJ
ajst-9878	11	36	problems	problem	NOUN
ajst-9878	11	37	,	,	PUNCT
ajst-9878	11	38	students	student	NOUN
ajst-9878	11	39	can	can	AUX
ajst-9878	11	40	gain	gain	VERB
ajst-9878	11	41	a	a	DET
ajst-9878	11	42	deeper	deep	ADJ
ajst-9878	11	43	understanding	understanding	NOUN
ajst-9878	11	44	of	of	ADP
ajst-9878	11	45	the	the	DET
ajst-9878	11	46	corresponding	corresponding	ADJ
ajst-9878	11	47	knowledge	knowledge	NOUN
ajst-9878	11	48	.	.	PUNCT
ajst-9878	12	1	the	the	DET
ajst-9878	12	2	second	second	ADJ
ajst-9878	12	3	is	be	AUX
ajst-9878	12	4	the	the	DET
ajst-9878	12	5	idea	idea	NOUN
ajst-9878	12	6	of	of	ADP
ajst-9878	12	7	permutation	permutation	NOUN
ajst-9878	12	8	.	.	PUNCT
ajst-9878	13	1	second	second	ADJ
ajst-9878	13	2	is	be	AUX
ajst-9878	13	3	the	the	DET
ajst-9878	13	4	idea	idea	NOUN
ajst-9878	13	5	of	of	ADP
ajst-9878	13	6	permutation	permutation	NOUN
ajst-9878	13	7	,	,	PUNCT
ajst-9878	13	8	which	which	PRON
ajst-9878	13	9	is	be	AUX
ajst-9878	13	10	an	an	DET
ajst-9878	13	11	important	important	ADJ
ajst-9878	13	12	way	way	NOUN
ajst-9878	13	13	of	of	ADP
ajst-9878	13	14	thinking	thinking	NOUN
ajst-9878	13	15	in	in	ADP
ajst-9878	13	16	mathematics	mathematics	NOUN
ajst-9878	13	17	teaching	teaching	NOUN
ajst-9878	13	18	and	and	CCONJ
ajst-9878	13	19	learning	learning	NOUN
ajst-9878	13	20	.	.	PUNCT
ajst-9878	14	1	it	it	PRON
ajst-9878	14	2	is	be	AUX
ajst-9878	14	3	accustomed	accustomed	ADJ
ajst-9878	14	4	to	to	ADP
ajst-9878	14	5	using	use	VERB
ajst-9878	14	6	letters	letter	NOUN
ajst-9878	14	7	instead	instead	ADV
ajst-9878	14	8	of	of	ADP
ajst-9878	14	9	formulas	formula	NOUN
ajst-9878	14	10	and	and	CCONJ
ajst-9878	14	11	employing	employ	VERB
ajst-9878	14	12	simplification	simplification	NOUN
ajst-9878	14	13	to	to	PART
ajst-9878	14	14	drive	drive	VERB
ajst-9878	14	15	problems	problem	NOUN
ajst-9878	14	16	to	to	ADP
ajst-9878	14	17	greater	great	ADJ
ajst-9878	14	18	simplicity	simplicity	NOUN
ajst-9878	14	19	.	.	PUNCT
ajst-9878	15	1	third	third	ADJ
ajst-9878	15	2	is	be	AUX
ajst-9878	15	3	the	the	DET
ajst-9878	15	4	idea	idea	NOUN
ajst-9878	15	5	of	of	ADP
ajst-9878	15	6	combining	combine	VERB
ajst-9878	15	7	numbers	number	NOUN
ajst-9878	15	8	and	and	CCONJ
ajst-9878	15	9	rows	row	NOUN
ajst-9878	15	10	.	.	PUNCT
ajst-9878	16	1	the	the	DET
ajst-9878	16	2	idea	idea	NOUN
ajst-9878	16	3	of	of	ADP
ajst-9878	16	4	combining	combine	VERB
ajst-9878	16	5	numbers	number	NOUN
ajst-9878	16	6	and	and	CCONJ
ajst-9878	16	7	rows	row	NOUN
ajst-9878	16	8	refers	refer	VERB
ajst-9878	16	9	to	to	ADP
ajst-9878	16	10	the	the	DET
ajst-9878	16	11	analysis	analysis	NOUN
ajst-9878	16	12	and	and	CCONJ
ajst-9878	16	13	study	study	NOUN
ajst-9878	16	14	of	of	ADP
ajst-9878	16	15	mathematical	mathematical	ADJ
ajst-9878	16	16	topics	topic	NOUN
ajst-9878	16	17	using	use	VERB
ajst-9878	16	18	a	a	DET
ajst-9878	16	19	combination	combination	NOUN
ajst-9878	16	20	of	of	ADP
ajst-9878	16	21	numbers	number	NOUN
ajst-9878	16	22	and	and	CCONJ
ajst-9878	16	23	rows	row	NOUN
ajst-9878	16	24	,	,	PUNCT
ajst-9878	16	25	transforming	transform	VERB
ajst-9878	16	26	problems	problem	NOUN
ajst-9878	16	27	that	that	PRON
ajst-9878	16	28	exist	exist	VERB
ajst-9878	16	29	in	in	ADP
ajst-9878	16	30	graphs	graph	NOUN
ajst-9878	16	31	into	into	ADP
ajst-9878	16	32	quantitative	quantitative	ADJ
ajst-9878	16	33	relationships	relationship	NOUN
ajst-9878	16	34	and	and	CCONJ
ajst-9878	16	35	making	make	VERB
ajst-9878	16	36	abstract	abstract	ADJ
ajst-9878	16	37	mathematical	mathematical	ADJ
ajst-9878	16	38	problems	problem	NOUN
ajst-9878	16	39	concrete	concrete	ADJ
ajst-9878	16	40	.	.	PUNCT
ajst-9878	17	1	the	the	DET
ajst-9878	17	2	fourth	fourth	ADJ
ajst-9878	17	3	is	be	AUX
ajst-9878	17	4	the	the	DET
ajst-9878	17	5	idea	idea	NOUN
ajst-9878	17	6	of	of	ADP
ajst-9878	17	7	function	function	NOUN
ajst-9878	17	8	,	,	PUNCT
ajst-9878	17	9	which	which	PRON
ajst-9878	17	10	is	be	AUX
ajst-9878	17	11	based	base	VERB
ajst-9878	17	12	on	on	ADP
ajst-9878	17	13	the	the	DET
ajst-9878	17	14	question	question	NOUN
ajst-9878	17	15	set	set	NOUN
ajst-9878	17	16	,	,	PUNCT
ajst-9878	17	17	the	the	DET
ajst-9878	17	18	construction	construction	NOUN
ajst-9878	17	19	of	of	ADP
ajst-9878	17	20	the	the	DET
ajst-9878	17	21	appropriate	appropriate	ADJ
ajst-9878	17	22	function	function	NOUN
ajst-9878	17	23	,	,	PUNCT
ajst-9878	17	24	through	through	ADP
ajst-9878	17	25	the	the	DET
ajst-9878	17	26	exploration	exploration	NOUN
ajst-9878	17	27	of	of	ADP
ajst-9878	17	28	the	the	DET
ajst-9878	17	29	properties	property	NOUN
ajst-9878	17	30	of	of	ADP
ajst-9878	17	31	the	the	DET
ajst-9878	17	32	function	function	NOUN
ajst-9878	17	33	,	,	PUNCT
ajst-9878	17	34	the	the	DET
ajst-9878	17	35	use	use	NOUN
ajst-9878	17	36	of	of	ADP
ajst-9878	17	37	the	the	DET
ajst-9878	17	38	combination	combination	NOUN
ajst-9878	17	39	of	of	ADP
ajst-9878	17	40	number	number	NOUN
ajst-9878	17	41	and	and	CCONJ
ajst-9878	17	42	shape	shape	NOUN
ajst-9878	17	43	thinking	think	VERB
ajst-9878	17	44	to	to	PART
ajst-9878	17	45	transform	transform	VERB
ajst-9878	17	46	research	research	NOUN
ajst-9878	17	47	.	.	PUNCT
ajst-9878	18	1	finally	finally	ADV
ajst-9878	18	2	,	,	PUNCT
ajst-9878	18	3	there	there	PRON
ajst-9878	18	4	is	be	VERB
ajst-9878	18	5	the	the	DET
ajst-9878	18	6	idea	idea	NOUN
ajst-9878	18	7	of	of	ADP
ajst-9878	18	8	reduction	reduction	NOUN
ajst-9878	18	9	,	,	PUNCT
ajst-9878	18	10	which	which	PRON
ajst-9878	18	11	is	be	AUX
ajst-9878	18	12	the	the	DET
ajst-9878	18	13	simplification	simplification	NOUN
ajst-9878	18	14	of	of	ADP
ajst-9878	18	15	real	real	ADJ
ajst-9878	18	16	concrete	concrete	ADJ
ajst-9878	18	17	problems	problem	NOUN
ajst-9878	18	18	by	by	ADP
ajst-9878	18	19	means	mean	NOUN
ajst-9878	18	20	of	of	ADP
ajst-9878	18	21	numbers	number	NOUN
ajst-9878	18	22	.	.	PUNCT
ajst-9878	19	1	in	in	ADP
ajst-9878	19	2	the	the	DET
ajst-9878	19	3	process	process	NOUN
ajst-9878	19	4	of	of	ADP
ajst-9878	19	5	transformation	transformation	NOUN
ajst-9878	19	6	,	,	PUNCT
ajst-9878	19	7	unsolved	unsolved	ADJ
ajst-9878	19	8	problems	problem	NOUN
ajst-9878	19	9	are	be	AUX
ajst-9878	19	10	transformed	transform	VERB
ajst-9878	19	11	into	into	ADP
ajst-9878	19	12	simple	simple	ADJ
ajst-9878	19	13	and	and	CCONJ
ajst-9878	19	14	easy	easy	ADJ
ajst-9878	19	15	to	to	PART
ajst-9878	19	16	solve	solve	VERB
ajst-9878	19	17	problems	problem	NOUN
ajst-9878	19	18	,	,	PUNCT
ajst-9878	19	19	and	and	CCONJ
ajst-9878	19	20	the	the	DET
ajst-9878	19	21	answers	answer	NOUN
ajst-9878	19	22	to	to	ADP
ajst-9878	19	23	complex	complex	ADJ
ajst-9878	19	24	problems	problem	NOUN
ajst-9878	19	25	are	be	AUX
ajst-9878	19	26	then	then	ADV
ajst-9878	19	27	derived	derive	VERB
ajst-9878	19	28	.	.	PUNCT
ajst-9878	20	1	2	2	X
ajst-9878	20	2	.	.	X
ajst-9878	20	3	application	application	NOUN
ajst-9878	20	4	of	of	ADP
ajst-9878	20	5	the	the	DET
ajst-9878	20	6	properties	property	NOUN
ajst-9878	20	7	of	of	ADP
ajst-9878	20	8	mathematical	mathematical	ADJ
ajst-9878	20	9	inequalities	inequality	NOUN
ajst-9878	20	10	inequalities	inequality	NOUN
ajst-9878	20	11	have	have	VERB
ajst-9878	20	12	their	their	PRON
ajst-9878	20	13	own	own	ADJ
ajst-9878	20	14	unique	unique	ADJ
ajst-9878	20	15	properties	property	NOUN
ajst-9878	20	16	which	which	PRON
ajst-9878	20	17	can	can	AUX
ajst-9878	20	18	be	be	AUX
ajst-9878	20	19	used	use	VERB
ajst-9878	20	20	to	to	PART
ajst-9878	20	21	solve	solve	VERB
ajst-9878	20	22	inequality	inequality	NOUN
ajst-9878	20	23	problems	problem	NOUN
ajst-9878	20	24	and	and	CCONJ
ajst-9878	20	25	prove	prove	VERB
ajst-9878	20	26	inequality	inequality	NOUN
ajst-9878	20	27	relations	relation	NOUN
ajst-9878	20	28	.	.	PUNCT
ajst-9878	21	1	our	our	PRON
ajst-9878	21	2	textbook	textbook	NOUN
ajst-9878	21	3	gives	give	VERB
ajst-9878	21	4	us	we	PRON
ajst-9878	21	5	some	some	DET
ajst-9878	21	6	basic	basic	ADJ
ajst-9878	21	7	properties	property	NOUN
ajst-9878	21	8	of	of	ADP
ajst-9878	21	9	inequalities	inequality	NOUN
ajst-9878	21	10	from	from	ADP
ajst-9878	21	11	which	which	PRON
ajst-9878	21	12	we	we	PRON
ajst-9878	21	13	can	can	AUX
ajst-9878	21	14	derive	derive	VERB
ajst-9878	21	15	some	some	DET
ajst-9878	21	16	potential	potential	ADJ
ajst-9878	21	17	properties	property	NOUN
ajst-9878	21	18	of	of	ADP
ajst-9878	21	19	inequalities	inequality	NOUN
ajst-9878	21	20	.	.	PUNCT
ajst-9878	22	1	by	by	ADP
ajst-9878	22	2	understanding	understand	VERB
ajst-9878	22	3	the	the	DET
ajst-9878	22	4	basic	basic	ADJ
ajst-9878	22	5	properties	property	NOUN
ajst-9878	22	6	in	in	ADP
ajst-9878	22	7	the	the	DET
ajst-9878	22	8	textbook	textbook	NOUN
ajst-9878	22	9	and	and	CCONJ
ajst-9878	22	10	mastering	master	VERB
ajst-9878	22	11	the	the	DET
ajst-9878	22	12	conditions	condition	NOUN
ajst-9878	22	13	for	for	ADP
ajst-9878	22	14	their	their	PRON
ajst-9878	22	15	use	use	NOUN
ajst-9878	22	16	and	and	CCONJ
ajst-9878	22	17	the	the	DET
ajst-9878	22	18	process	process	NOUN
ajst-9878	22	19	of	of	ADP
ajst-9878	22	20	proof	proof	NOUN
ajst-9878	22	21	,	,	PUNCT
ajst-9878	22	22	we	we	PRON
ajst-9878	22	23	can	can	AUX
ajst-9878	22	24	grasp	grasp	VERB
ajst-9878	22	25	the	the	DET
ajst-9878	22	26	close	close	ADJ
ajst-9878	22	27	connection	connection	NOUN
ajst-9878	22	28	between	between	ADP
ajst-9878	22	29	each	each	DET
ajst-9878	22	30	property	property	NOUN
ajst-9878	22	31	and	and	CCONJ
ajst-9878	22	32	thus	thus	ADV
ajst-9878	22	33	use	use	VERB
ajst-9878	22	34	inequalities	inequality	NOUN
ajst-9878	22	35	flexibly	flexibly	ADV
ajst-9878	22	36	to	to	PART
ajst-9878	22	37	solve	solve	VERB
ajst-9878	22	38	problems	problem	NOUN
ajst-9878	22	39	.	.	PUNCT
ajst-9878	23	1	(	(	PUNCT
ajst-9878	23	2	a	a	X
ajst-9878	23	3	)	)	PUNCT
ajst-9878	23	4	the	the	DET
ajst-9878	23	5	conditions	condition	NOUN
ajst-9878	23	6	under	under	ADP
ajst-9878	23	7	which	which	PRON
ajst-9878	23	8	the	the	DET
ajst-9878	23	9	properties	property	NOUN
ajst-9878	23	10	of	of	ADP
ajst-9878	23	11	inequalities	inequality	NOUN
ajst-9878	23	12	hold	hold	VERB
ajst-9878	23	13	.	.	PUNCT
ajst-9878	24	1	when	when	SCONJ
ajst-9878	24	2	we	we	PRON
ajst-9878	24	3	use	use	VERB
ajst-9878	24	4	the	the	DET
ajst-9878	24	5	properties	property	NOUN
ajst-9878	24	6	of	of	ADP
ajst-9878	24	7	inequalities	inequality	NOUN
ajst-9878	24	8	to	to	PART
ajst-9878	24	9	answer	answer	VERB
ajst-9878	24	10	some	some	DET
ajst-9878	24	11	inequality	inequality	NOUN
ajst-9878	24	12	questions	question	NOUN
ajst-9878	24	13	we	we	PRON
ajst-9878	24	14	must	must	AUX
ajst-9878	24	15	be	be	AUX
ajst-9878	24	16	skilled	skilled	ADJ
ajst-9878	24	17	in	in	ADP
ajst-9878	24	18	the	the	DET
ajst-9878	24	19	conditions	condition	NOUN
ajst-9878	24	20	under	under	ADP
ajst-9878	24	21	which	which	PRON
ajst-9878	24	22	inequalities	inequality	NOUN
ajst-9878	24	23	are	be	AUX
ajst-9878	24	24	established	establish	VERB
ajst-9878	24	25	or	or	CCONJ
ajst-9878	24	26	there	there	PRON
ajst-9878	24	27	will	will	AUX
ajst-9878	24	28	be	be	AUX
ajst-9878	24	29	certain	certain	ADJ
ajst-9878	24	30	errors	error	NOUN
ajst-9878	24	31	in	in	ADP
ajst-9878	24	32	the	the	DET
ajst-9878	24	33	process	process	NOUN
ajst-9878	24	34	of	of	ADP
ajst-9878	24	35	application	application	NOUN
ajst-9878	24	36	.	.	PUNCT
ajst-9878	25	1	the	the	DET
ajst-9878	25	2	arrows	arrow	NOUN
ajst-9878	25	3	used	use	VERB
ajst-9878	25	4	to	to	PART
ajst-9878	25	5	represent	represent	VERB
ajst-9878	25	6	the	the	DET
ajst-9878	25	7	properties	property	NOUN
ajst-9878	25	8	of	of	ADP
ajst-9878	25	9	inequalities	inequality	NOUN
ajst-9878	25	10	should	should	AUX
ajst-9878	25	11	be	be	AUX
ajst-9878	25	12	clearly	clearly	ADV
ajst-9878	25	13	seen	see	VERB
ajst-9878	25	14	to	to	PART
ajst-9878	25	15	note	note	VERB
ajst-9878	25	16	whether	whether	SCONJ
ajst-9878	25	17	they	they	PRON
ajst-9878	25	18	are	be	AUX
ajst-9878	25	19	unidirectional	unidirectional	ADJ
ajst-9878	25	20	or	or	CCONJ
ajst-9878	25	21	bidirectional	bidirectional	NOUN
ajst-9878	25	22	,	,	PUNCT
ajst-9878	25	23	which	which	PRON
ajst-9878	25	24	simply	simply	ADV
ajst-9878	25	25	means	mean	VERB
ajst-9878	25	26	that	that	SCONJ
ajst-9878	25	27	each	each	DET
ajst-9878	25	28	property	property	NOUN
ajst-9878	25	29	should	should	AUX
ajst-9878	25	30	be	be	AUX
ajst-9878	25	31	determined	determine	VERB
ajst-9878	25	32	to	to	PART
ajst-9878	25	33	be	be	AUX
ajst-9878	25	34	reversible	reversible	ADJ
ajst-9878	25	35	.	.	PUNCT
ajst-9878	26	1	(	(	PUNCT
ajst-9878	26	2	b	b	X
ajst-9878	26	3	)	)	PUNCT
ajst-9878	26	4	use	use	VERB
ajst-9878	26	5	the	the	DET
ajst-9878	26	6	properties	property	NOUN
ajst-9878	26	7	of	of	ADP
ajst-9878	26	8	inequalities	inequality	NOUN
ajst-9878	26	9	to	to	PART
ajst-9878	26	10	prove	prove	VERB
ajst-9878	26	11	inequalities	inequality	NOUN
ajst-9878	26	12	.	.	PUNCT
ajst-9878	27	1	using	use	VERB
ajst-9878	27	2	the	the	DET
ajst-9878	27	3	basic	basic	ADJ
ajst-9878	27	4	properties	property	NOUN
ajst-9878	27	5	of	of	ADP
ajst-9878	27	6	inequalities	inequality	NOUN
ajst-9878	27	7	and	and	CCONJ
ajst-9878	27	8	some	some	PRON
ajst-9878	27	9	of	of	ADP
ajst-9878	27	10	the	the	DET
ajst-9878	27	11	properties	property	NOUN
ajst-9878	27	12	derived	derive	VERB
ajst-9878	27	13	through	through	ADP
ajst-9878	27	14	derivation	derivation	NOUN
ajst-9878	27	15	we	we	PRON
ajst-9878	27	16	can	can	AUX
ajst-9878	27	17	prove	prove	VERB
ajst-9878	27	18	some	some	DET
ajst-9878	27	19	inequality	inequality	NOUN
ajst-9878	27	20	problems	problem	NOUN
ajst-9878	27	21	.	.	PUNCT
ajst-9878	28	1	the	the	DET
ajst-9878	28	2	principle	principle	NOUN
ajst-9878	28	3	we	we	PRON
ajst-9878	28	4	must	must	AUX
ajst-9878	28	5	follow	follow	VERB
ajst-9878	28	6	in	in	ADP
ajst-9878	28	7	solving	solve	VERB
ajst-9878	28	8	inequality	inequality	NOUN
ajst-9878	28	9	problems	problem	NOUN
ajst-9878	28	10	is	be	AUX
ajst-9878	28	11	to	to	PART
ajst-9878	28	12	understand	understand	VERB
ajst-9878	28	13	these	these	DET
ajst-9878	28	14	properties	property	NOUN
ajst-9878	28	15	on	on	ADP
ajst-9878	28	16	the	the	DET
ajst-9878	28	17	basis	basis	NOUN
ajst-9878	28	18	of	of	ADP
ajst-9878	28	19	skillful	skillful	ADJ
ajst-9878	28	20	and	and	CCONJ
ajst-9878	28	21	flexible	flexible	ADJ
ajst-9878	28	22	use	use	NOUN
ajst-9878	28	23	so	so	SCONJ
ajst-9878	28	24	that	that	SCONJ
ajst-9878	28	25	we	we	PRON
ajst-9878	28	26	can	can	AUX
ajst-9878	28	27	accurately	accurately	ADV
ajst-9878	28	28	answer	answer	VERB
ajst-9878	28	29	the	the	DET
ajst-9878	28	30	questions	question	NOUN
ajst-9878	28	31	.	.	PUNCT
ajst-9878	29	1	(	(	PUNCT
ajst-9878	29	2	c	c	X
ajst-9878	29	3	)	)	PUNCT
ajst-9878	29	4	use	use	VERB
ajst-9878	29	5	the	the	DET
ajst-9878	29	6	properties	property	NOUN
ajst-9878	29	7	of	of	ADP
ajst-9878	29	8	inequalities	inequality	NOUN
ajst-9878	29	9	to	to	PART
ajst-9878	29	10	find	find	VERB
ajst-9878	29	11	the	the	DET
ajst-9878	29	12	range	range	NOUN
ajst-9878	29	13	.	.	PUNCT
ajst-9878	30	1	in	in	ADP
ajst-9878	30	2	the	the	DET
ajst-9878	30	3	actual	actual	ADJ
ajst-9878	30	4	learning	learning	NOUN
ajst-9878	30	5	process	process	NOUN
ajst-9878	30	6	we	we	PRON
ajst-9878	30	7	often	often	ADV
ajst-9878	30	8	encounter	encounter	VERB
ajst-9878	30	9	some	some	DET
ajst-9878	30	10	problems	problem	NOUN
ajst-9878	30	11	that	that	PRON
ajst-9878	30	12	require	require	VERB
ajst-9878	30	13	a	a	DET
ajst-9878	30	14	specific	specific	ADJ
ajst-9878	30	15	range	range	NOUN
ajst-9878	30	16	of	of	ADP
ajst-9878	30	17	inequalities	inequality	NOUN
ajst-9878	30	18	then	then	ADV
ajst-9878	30	19	we	we	PRON
ajst-9878	30	20	can	can	AUX
ajst-9878	30	21	use	use	VERB
ajst-9878	30	22	the	the	DET
ajst-9878	30	23	range	range	NOUN
ajst-9878	30	24	of	of	ADP
ajst-9878	30	25	several	several	ADJ
ajst-9878	30	26	inequalities	inequality	NOUN
ajst-9878	30	27	combined	combine	VERB
ajst-9878	30	28	to	to	PART
ajst-9878	30	29	solve	solve	VERB
ajst-9878	30	30	them	they	PRON
ajst-9878	30	31	.	.	PUNCT
ajst-9878	31	1	in	in	ADP
ajst-9878	31	2	answering	answer	VERB
ajst-9878	31	3	such	such	ADJ
ajst-9878	31	4	questions	question	NOUN
ajst-9878	31	5	we	we	PRON
ajst-9878	31	6	should	should	AUX
ajst-9878	31	7	note	note	VERB
ajst-9878	31	8	that	that	SCONJ
ajst-9878	31	9	"	"	PUNCT
ajst-9878	31	10	the	the	DET
ajst-9878	31	11	two	two	NUM
ajst-9878	31	12	sides	side	NOUN
ajst-9878	31	13	of	of	ADP
ajst-9878	31	14	an	an	DET
ajst-9878	31	15	inequality	inequality	NOUN
ajst-9878	31	16	can	can	AUX
ajst-9878	31	17	be	be	AUX
ajst-9878	31	18	added	add	VERB
ajst-9878	31	19	together	together	ADV
ajst-9878	31	20	(	(	PUNCT
ajst-9878	31	21	subtracted	subtract	VERB
ajst-9878	31	22	)	)	PUNCT
ajst-9878	31	23	"	"	PUNCT
ajst-9878	31	24	but	but	CCONJ
ajst-9878	31	25	this	this	DET
ajst-9878	31	26	transformation	transformation	NOUN
ajst-9878	31	27	is	be	AUX
ajst-9878	31	28	not	not	PART
ajst-9878	31	29	an	an	DET
ajst-9878	31	30	equivalence	equivalence	NOUN
ajst-9878	31	31	deformation	deformation	NOUN
ajst-9878	31	32	plus	plus	CCONJ
ajst-9878	31	33	the	the	DET
ajst-9878	31	34	repeated	repeat	VERB
ajst-9878	31	35	use	use	NOUN
ajst-9878	31	36	of	of	ADP
ajst-9878	31	37	this	this	DET
ajst-9878	31	38	transformation	transformation	NOUN
ajst-9878	31	39	in	in	ADP
ajst-9878	31	40	the	the	DET
ajst-9878	31	41	process	process	NOUN
ajst-9878	31	42	of	of	ADP
ajst-9878	31	43	answering	answer	VERB
ajst-9878	31	44	a	a	DET
ajst-9878	31	45	question	question	NOUN
ajst-9878	31	46	may	may	AUX
ajst-9878	31	47	invariably	invariably	ADV
ajst-9878	31	48	increase	increase	VERB
ajst-9878	31	49	the	the	DET
ajst-9878	31	50	true	true	ADJ
ajst-9878	31	51	range	range	NOUN
ajst-9878	31	52	of	of	ADP
ajst-9878	31	53	values	value	NOUN
ajst-9878	31	54	and	and	CCONJ
ajst-9878	31	55	thus	thus	ADV
ajst-9878	31	56	make	make	VERB
ajst-9878	31	57	the	the	DET
ajst-9878	31	58	calculation	calculation	NOUN
ajst-9878	31	59	wrong	wrong	ADJ
ajst-9878	31	60	.	.	PUNCT
ajst-9878	32	1	by	by	ADP
ajst-9878	32	2	first	first	ADV
ajst-9878	32	3	establishing	establish	VERB
ajst-9878	32	4	the	the	DET
ajst-9878	32	5	equivalence	equivalence	NOUN
ajst-9878	32	6	of	of	ADP
ajst-9878	32	7	the	the	DET
ajst-9878	32	8	whole	whole	NOUN
ajst-9878	32	9	of	of	ADP
ajst-9878	32	10	the	the	DET
ajst-9878	32	11	range	range	NOUN
ajst-9878	32	12	to	to	PART
ajst-9878	32	13	be	be	AUX
ajst-9878	32	14	sought	seek	VERB
ajst-9878	32	15	and	and	CCONJ
ajst-9878	32	16	the	the	DET
ajst-9878	32	17	whole	whole	NOUN
ajst-9878	32	18	of	of	ADP
ajst-9878	32	19	the	the	DET
ajst-9878	32	20	known	known	ADJ
ajst-9878	32	21	range	range	NOUN
ajst-9878	32	22	,	,	PUNCT
ajst-9878	32	23	and	and	CCONJ
ajst-9878	32	24	finally	finally	ADV
ajst-9878	32	25	by	by	ADP
ajst-9878	32	26	using	use	VERB
ajst-9878	32	27	the	the	DET
ajst-9878	32	28	"	"	PUNCT
ajst-9878	32	29	one	one	NUM
ajst-9878	32	30	-	-	PUNCT
ajst-9878	32	31	time	time	NOUN
ajst-9878	32	32	inequality	inequality	NOUN
ajst-9878	32	33	operation	operation	NOUN
ajst-9878	32	34	to	to	PART
ajst-9878	32	35	find	find	VERB
ajst-9878	32	36	the	the	DET
ajst-9878	32	37	range	range	NOUN
ajst-9878	32	38	to	to	PART
ajst-9878	32	39	be	be	AUX
ajst-9878	32	40	sought	seek	VERB
ajst-9878	32	41	"	"	PUNCT
ajst-9878	32	42	method	method	NOUN
ajst-9878	32	43	,	,	PUNCT
ajst-9878	32	44	we	we	PRON
ajst-9878	32	45	will	will	AUX
ajst-9878	32	46	avoid	avoid	VERB
ajst-9878	32	47	unnecessary	unnecessary	ADJ
ajst-9878	32	48	errors	error	NOUN
ajst-9878	32	49	.	.	PUNCT
ajst-9878	33	1	3	3	X
ajst-9878	33	2	.	.	X
ajst-9878	33	3	learning	learn	VERB
ajst-9878	33	4	strategies	strategy	NOUN
ajst-9878	33	5	for	for	ADP
ajst-9878	33	6	mathematical	mathematical	ADJ
ajst-9878	33	7	inequalities	inequality	NOUN
ajst-9878	33	8	by	by	ADP
ajst-9878	33	9	analysing	analyse	VERB
ajst-9878	33	10	and	and	CCONJ
ajst-9878	33	11	summarising	summarise	VERB
ajst-9878	33	12	the	the	DET
ajst-9878	33	13	relevant	relevant	ADJ
ajst-9878	33	14	mathematical	mathematical	ADJ
ajst-9878	33	15	theories	theory	NOUN
ajst-9878	33	16	of	of	ADP
ajst-9878	33	17	education	education	NOUN
ajst-9878	33	18	and	and	CCONJ
ajst-9878	33	19	the	the	DET
ajst-9878	33	20	basic	basic	ADJ
ajst-9878	33	21	content	content	NOUN
ajst-9878	33	22	of	of	ADP
ajst-9878	33	23	inequalities	inequality	NOUN
ajst-9878	33	24	that	that	PRON
ajst-9878	33	25	we	we	PRON
ajst-9878	33	26	usually	usually	ADV
ajst-9878	33	27	learn	learn	VERB
ajst-9878	33	28	we	we	PRON
ajst-9878	33	29	can	can	AUX
ajst-9878	33	30	draw	draw	VERB
ajst-9878	33	31	some	some	DET
ajst-9878	33	32	corresponding	corresponding	ADJ
ajst-9878	33	33	conclusions	conclusion	NOUN
ajst-9878	33	34	.	.	PUNCT
ajst-9878	34	1	our	our	PRON
ajst-9878	34	2	learning	learning	NOUN
ajst-9878	34	3	process	process	NOUN
ajst-9878	34	4	is	be	AUX
ajst-9878	34	5	in	in	ADP
ajst-9878	34	6	fact	fact	NOUN
ajst-9878	34	7	a	a	DET
ajst-9878	34	8	process	process	NOUN
ajst-9878	34	9	of	of	ADP
ajst-9878	34	10	communication	communication	NOUN
ajst-9878	34	11	,	,	PUNCT
ajst-9878	34	12	understanding	understanding	NOUN
ajst-9878	34	13	and	and	CCONJ
ajst-9878	34	14	innovation	innovation	NOUN
ajst-9878	34	15	.	.	PUNCT
ajst-9878	35	1	when	when	SCONJ
ajst-9878	35	2	we	we	PRON
ajst-9878	35	3	are	be	AUX
ajst-9878	35	4	165	165	NUM
ajst-9878	35	5	learning	learning	NOUN
ajst-9878	35	6	,	,	PUNCT
ajst-9878	35	7	we	we	PRON
ajst-9878	35	8	can	can	AUX
ajst-9878	35	9	not	not	PART
ajst-9878	35	10	just	just	ADV
ajst-9878	35	11	memorise	memorise	VERB
ajst-9878	35	12	what	what	PRON
ajst-9878	35	13	the	the	DET
ajst-9878	35	14	teacher	teacher	NOUN
ajst-9878	35	15	says	say	VERB
ajst-9878	35	16	in	in	ADP
ajst-9878	35	17	our	our	PRON
ajst-9878	35	18	heads	head	NOUN
ajst-9878	35	19	,	,	PUNCT
ajst-9878	35	20	but	but	CCONJ
ajst-9878	35	21	we	we	PRON
ajst-9878	35	22	must	must	AUX
ajst-9878	35	23	constantly	constantly	ADV
ajst-9878	35	24	analyse	analyse	VERB
ajst-9878	35	25	and	and	CCONJ
ajst-9878	35	26	think	think	VERB
ajst-9878	35	27	about	about	ADP
ajst-9878	35	28	the	the	DET
ajst-9878	35	29	problems	problem	NOUN
ajst-9878	35	30	and	and	CCONJ
ajst-9878	35	31	fully	fully	ADV
ajst-9878	35	32	open	open	VERB
ajst-9878	35	33	our	our	PRON
ajst-9878	35	34	brains	brain	NOUN
ajst-9878	35	35	to	to	PART
ajst-9878	35	36	combine	combine	VERB
ajst-9878	35	37	the	the	DET
ajst-9878	35	38	theoretical	theoretical	ADJ
ajst-9878	35	39	knowledge	knowledge	NOUN
ajst-9878	35	40	and	and	CCONJ
ajst-9878	35	41	solutions	solution	NOUN
ajst-9878	35	42	we	we	PRON
ajst-9878	35	43	have	have	AUX
ajst-9878	35	44	learned	learn	VERB
ajst-9878	35	45	with	with	ADP
ajst-9878	35	46	the	the	DET
ajst-9878	35	47	practical	practical	ADJ
ajst-9878	35	48	problems	problem	NOUN
ajst-9878	35	49	,	,	PUNCT
ajst-9878	35	50	so	so	SCONJ
ajst-9878	35	51	that	that	SCONJ
ajst-9878	35	52	we	we	PRON
ajst-9878	35	53	can	can	AUX
ajst-9878	35	54	learn	learn	VERB
ajst-9878	35	55	better	well	ADV
ajst-9878	35	56	.	.	PUNCT
ajst-9878	36	1	1	1	X
ajst-9878	36	2	.	.	X
ajst-9878	36	3	understanding	understanding	NOUN
ajst-9878	36	4	and	and	CCONJ
ajst-9878	36	5	mastering	master	VERB
ajst-9878	36	6	mathematical	mathematical	ADJ
ajst-9878	36	7	thinking	thinking	NOUN
ajst-9878	36	8	in	in	ADP
ajst-9878	36	9	the	the	DET
ajst-9878	36	10	process	process	NOUN
ajst-9878	36	11	of	of	ADP
ajst-9878	36	12	learning	learning	NOUN
ajst-9878	36	13	,	,	PUNCT
ajst-9878	36	14	we	we	PRON
ajst-9878	36	15	can	can	AUX
ajst-9878	36	16	design	design	VERB
ajst-9878	36	17	our	our	PRON
ajst-9878	36	18	own	own	ADJ
ajst-9878	36	19	real	real	ADJ
ajst-9878	36	20	-	-	PUNCT
ajst-9878	36	21	life	life	NOUN
ajst-9878	36	22	situations	situation	NOUN
ajst-9878	36	23	to	to	PART
ajst-9878	36	24	link	link	VERB
ajst-9878	36	25	our	our	PRON
ajst-9878	36	26	knowledge	knowledge	NOUN
ajst-9878	36	27	of	of	ADP
ajst-9878	36	28	inequalities	inequality	NOUN
ajst-9878	36	29	.	.	PUNCT
ajst-9878	37	1	in	in	ADP
ajst-9878	37	2	fact	fact	NOUN
ajst-9878	37	3	,	,	PUNCT
ajst-9878	37	4	mathematical	mathematical	ADJ
ajst-9878	37	5	knowledge	knowledge	NOUN
ajst-9878	37	6	is	be	AUX
ajst-9878	37	7	systematic	systematic	ADJ
ajst-9878	37	8	and	and	CCONJ
ajst-9878	37	9	coherent	coherent	ADJ
ajst-9878	37	10	.	.	PUNCT
ajst-9878	38	1	the	the	DET
ajst-9878	38	2	inequalities	inequality	NOUN
ajst-9878	38	3	we	we	PRON
ajst-9878	38	4	are	be	AUX
ajst-9878	38	5	learning	learn	VERB
ajst-9878	38	6	now	now	ADV
ajst-9878	38	7	are	be	AUX
ajst-9878	38	8	actually	actually	ADV
ajst-9878	38	9	a	a	DET
ajst-9878	38	10	supplement	supplement	NOUN
ajst-9878	38	11	and	and	CCONJ
ajst-9878	38	12	extension	extension	NOUN
ajst-9878	38	13	of	of	ADP
ajst-9878	38	14	our	our	PRON
ajst-9878	38	15	knowledge	knowledge	NOUN
ajst-9878	38	16	in	in	ADP
ajst-9878	38	17	junior	junior	ADJ
ajst-9878	38	18	secondary	secondary	ADJ
ajst-9878	38	19	school	school	NOUN
ajst-9878	38	20	,	,	PUNCT
ajst-9878	38	21	and	and	CCONJ
ajst-9878	38	22	a	a	DET
ajst-9878	38	23	refinement	refinement	NOUN
ajst-9878	38	24	and	and	CCONJ
ajst-9878	38	25	enhancement	enhancement	NOUN
ajst-9878	38	26	of	of	ADP
ajst-9878	38	27	our	our	PRON
ajst-9878	38	28	previous	previous	ADJ
ajst-9878	38	29	knowledge	knowledge	NOUN
ajst-9878	38	30	.	.	PUNCT
ajst-9878	39	1	therefore	therefore	ADV
ajst-9878	39	2	,	,	PUNCT
ajst-9878	39	3	we	we	PRON
ajst-9878	39	4	should	should	AUX
ajst-9878	39	5	learn	learn	VERB
ajst-9878	39	6	to	to	PART
ajst-9878	39	7	summarise	summarise	VERB
ajst-9878	39	8	our	our	PRON
ajst-9878	39	9	knowledge	knowledge	NOUN
ajst-9878	39	10	of	of	ADP
ajst-9878	39	11	inequalities	inequality	NOUN
ajst-9878	39	12	at	at	ADP
ajst-9878	39	13	a	a	DET
ajst-9878	39	14	deeper	deep	ADJ
ajst-9878	39	15	level	level	NOUN
ajst-9878	39	16	in	in	ADP
ajst-9878	39	17	order	order	NOUN
ajst-9878	39	18	to	to	PART
ajst-9878	39	19	improve	improve	VERB
ajst-9878	39	20	our	our	PRON
ajst-9878	39	21	cognitive	cognitive	ADJ
ajst-9878	39	22	ability	ability	NOUN
ajst-9878	39	23	.	.	PUNCT
ajst-9878	40	1	2	2	X
ajst-9878	40	2	.	.	X
ajst-9878	40	3	focus	focus	VERB
ajst-9878	40	4	on	on	ADP
ajst-9878	40	5	the	the	DET
ajst-9878	40	6	exploration	exploration	NOUN
ajst-9878	40	7	of	of	ADP
ajst-9878	40	8	inequality	inequality	NOUN
ajst-9878	40	9	solutions	solution	NOUN
ajst-9878	40	10	,	,	PUNCT
ajst-9878	40	11	improve	improve	VERB
ajst-9878	40	12	thinking	thinking	NOUN
ajst-9878	40	13	skills	skill	NOUN
ajst-9878	40	14	and	and	CCONJ
ajst-9878	40	15	enhance	enhance	VERB
ajst-9878	40	16	the	the	DET
ajst-9878	40	17	connection	connection	NOUN
ajst-9878	40	18	between	between	ADP
ajst-9878	40	19	knowledge	knowledge	NOUN
ajst-9878	40	20	.	.	PUNCT
ajst-9878	41	1	through	through	ADP
ajst-9878	41	2	our	our	PRON
ajst-9878	41	3	study	study	NOUN
ajst-9878	41	4	,	,	PUNCT
ajst-9878	41	5	we	we	PRON
ajst-9878	41	6	can	can	AUX
ajst-9878	41	7	know	know	VERB
ajst-9878	41	8	the	the	DET
ajst-9878	41	9	properties	property	NOUN
ajst-9878	41	10	of	of	ADP
ajst-9878	41	11	inequalities	inequality	NOUN
ajst-9878	41	12	and	and	CCONJ
ajst-9878	41	13	inequalities	inequality	NOUN
ajst-9878	41	14	are	be	AUX
ajst-9878	41	15	the	the	DET
ajst-9878	41	16	basis	basis	NOUN
ajst-9878	41	17	of	of	ADP
ajst-9878	41	18	the	the	DET
ajst-9878	41	19	knowledge	knowledge	NOUN
ajst-9878	41	20	of	of	ADP
ajst-9878	41	21	inequalities	inequality	NOUN
ajst-9878	41	22	and	and	CCONJ
ajst-9878	41	23	the	the	DET
ajst-9878	41	24	solution	solution	NOUN
ajst-9878	41	25	of	of	ADP
ajst-9878	41	26	inequalities	inequality	NOUN
ajst-9878	41	27	requires	require	VERB
ajst-9878	41	28	us	we	PRON
ajst-9878	41	29	to	to	PART
ajst-9878	41	30	have	have	VERB
ajst-9878	41	31	strong	strong	ADJ
ajst-9878	41	32	arithmetic	arithmetic	ADJ
ajst-9878	41	33	skills	skill	NOUN
ajst-9878	41	34	,	,	PUNCT
ajst-9878	41	35	so	so	SCONJ
ajst-9878	41	36	that	that	SCONJ
ajst-9878	41	37	we	we	PRON
ajst-9878	41	38	can	can	AUX
ajst-9878	41	39	better	well	ADV
ajst-9878	41	40	use	use	VERB
ajst-9878	41	41	and	and	CCONJ
ajst-9878	41	42	transfer	transfer	VERB
ajst-9878	41	43	the	the	DET
ajst-9878	41	44	mathematical	mathematical	ADJ
ajst-9878	41	45	knowledge	knowledge	NOUN
ajst-9878	41	46	we	we	PRON
ajst-9878	41	47	have	have	AUX
ajst-9878	41	48	learnt	learn	VERB
ajst-9878	41	49	and	and	CCONJ
ajst-9878	41	50	innovate	innovate	VERB
ajst-9878	41	51	.	.	PUNCT
ajst-9878	42	1	in	in	ADP
ajst-9878	42	2	general	general	ADJ
ajst-9878	42	3	,	,	PUNCT
ajst-9878	42	4	we	we	PRON
ajst-9878	42	5	should	should	AUX
ajst-9878	42	6	pay	pay	VERB
ajst-9878	42	7	attention	attention	NOUN
ajst-9878	42	8	to	to	ADP
ajst-9878	42	9	the	the	DET
ajst-9878	42	10	learning	learning	NOUN
ajst-9878	42	11	of	of	ADP
ajst-9878	42	12	inequalities	inequality	NOUN
ajst-9878	42	13	with	with	ADP
ajst-9878	42	14	parameters	parameter	NOUN
ajst-9878	42	15	.	.	PUNCT
ajst-9878	43	1	when	when	SCONJ
ajst-9878	43	2	learning	learn	VERB
ajst-9878	43	3	the	the	DET
ajst-9878	43	4	whole	whole	ADJ
ajst-9878	43	5	system	system	NOUN
ajst-9878	43	6	of	of	ADP
ajst-9878	43	7	inequalities	inequality	NOUN
ajst-9878	43	8	,	,	PUNCT
ajst-9878	43	9	we	we	PRON
ajst-9878	43	10	must	must	AUX
ajst-9878	43	11	not	not	PART
ajst-9878	43	12	learn	learn	VERB
ajst-9878	43	13	them	they	PRON
ajst-9878	43	14	in	in	ADP
ajst-9878	43	15	isolation	isolation	NOUN
ajst-9878	43	16	,	,	PUNCT
ajst-9878	43	17	but	but	CCONJ
ajst-9878	43	18	in	in	ADP
ajst-9878	43	19	the	the	DET
ajst-9878	43	20	context	context	NOUN
ajst-9878	43	21	of	of	ADP
ajst-9878	43	22	mathematics	mathematic	NOUN
ajst-9878	43	23	.	.	PUNCT
ajst-9878	44	1	3	3	X
ajst-9878	44	2	.	.	X
ajst-9878	44	3	through	through	ADP
ajst-9878	44	4	the	the	DET
ajst-9878	44	5	observation	observation	NOUN
ajst-9878	44	6	of	of	ADP
ajst-9878	44	7	the	the	DET
ajst-9878	44	8	reasoning	reasoning	NOUN
ajst-9878	44	9	and	and	CCONJ
ajst-9878	44	10	argumentative	argumentative	ADJ
ajst-9878	44	11	process	process	NOUN
ajst-9878	44	12	,	,	PUNCT
ajst-9878	44	13	to	to	PART
ajst-9878	44	14	develop	develop	VERB
ajst-9878	44	15	their	their	PRON
ajst-9878	44	16	own	own	ADJ
ajst-9878	44	17	abstract	abstract	ADJ
ajst-9878	44	18	thinking	thinking	NOUN
ajst-9878	44	19	skills	skill	NOUN
ajst-9878	44	20	.	.	PUNCT
ajst-9878	45	1	when	when	SCONJ
ajst-9878	45	2	we	we	PRON
ajst-9878	45	3	study	study	VERB
ajst-9878	45	4	inequalities	inequality	NOUN
ajst-9878	45	5	,	,	PUNCT
ajst-9878	45	6	we	we	PRON
ajst-9878	45	7	should	should	AUX
ajst-9878	45	8	observe	observe	VERB
ajst-9878	45	9	the	the	DET
ajst-9878	45	10	reasoning	reasoning	NOUN
ajst-9878	45	11	and	and	CCONJ
ajst-9878	45	12	argumentation	argumentation	NOUN
ajst-9878	45	13	process	process	NOUN
ajst-9878	45	14	.	.	PUNCT
ajst-9878	46	1	through	through	ADP
ajst-9878	46	2	the	the	DET
ajst-9878	46	3	study	study	NOUN
ajst-9878	46	4	of	of	ADP
ajst-9878	46	5	basic	basic	ADJ
ajst-9878	46	6	inequalities	inequality	NOUN
ajst-9878	46	7	,	,	PUNCT
ajst-9878	46	8	we	we	PRON
ajst-9878	46	9	can	can	AUX
ajst-9878	46	10	appreciate	appreciate	VERB
ajst-9878	46	11	the	the	DET
ajst-9878	46	12	combination	combination	NOUN
ajst-9878	46	13	of	of	ADP
ajst-9878	46	14	number	number	NOUN
ajst-9878	46	15	and	and	CCONJ
ajst-9878	46	16	shape	shape	NOUN
ajst-9878	46	17	and	and	CCONJ
ajst-9878	46	18	other	other	ADJ
ajst-9878	46	19	methods	method	NOUN
ajst-9878	46	20	of	of	ADP
ajst-9878	46	21	thought	thought	NOUN
ajst-9878	46	22	,	,	PUNCT
ajst-9878	46	23	thus	thus	ADV
ajst-9878	46	24	improving	improve	VERB
ajst-9878	46	25	our	our	PRON
ajst-9878	46	26	logical	logical	ADJ
ajst-9878	46	27	and	and	CCONJ
ajst-9878	46	28	abstract	abstract	ADJ
ajst-9878	46	29	thinking	thinking	NOUN
ajst-9878	46	30	skills	skill	NOUN
ajst-9878	46	31	and	and	CCONJ
ajst-9878	46	32	developing	develop	VERB
ajst-9878	46	33	a	a	DET
ajst-9878	46	34	rigorous	rigorous	ADJ
ajst-9878	46	35	and	and	CCONJ
ajst-9878	46	36	standardised	standardised	ADJ
ajst-9878	46	37	learning	learning	NOUN
ajst-9878	46	38	ability	ability	NOUN
ajst-9878	46	39	and	and	CCONJ
ajst-9878	46	40	the	the	DET
ajst-9878	46	41	ability	ability	NOUN
ajst-9878	46	42	to	to	PART
ajst-9878	46	43	analyse	analyse	VERB
ajst-9878	46	44	and	and	CCONJ
ajst-9878	46	45	solve	solve	VERB
ajst-9878	46	46	problems	problem	NOUN
ajst-9878	46	47	.	.	PUNCT
ajst-9878	47	1	4	4	X
ajst-9878	47	2	.	.	X
ajst-9878	47	3	strengthen	strengthen	VERB
ajst-9878	47	4	the	the	DET
ajst-9878	47	5	connection	connection	NOUN
ajst-9878	47	6	of	of	ADP
ajst-9878	47	7	knowledge	knowledge	NOUN
ajst-9878	47	8	,	,	PUNCT
ajst-9878	47	9	the	the	DET
ajst-9878	47	10	actual	actual	ADJ
ajst-9878	47	11	life	life	NOUN
ajst-9878	47	12	of	of	ADP
ajst-9878	47	13	the	the	DET
ajst-9878	47	14	problem	problem	NOUN
ajst-9878	47	15	of	of	ADP
ajst-9878	47	16	mathematical	mathematical	ADJ
ajst-9878	47	17	abstraction	abstraction	NOUN
ajst-9878	47	18	.	.	PUNCT
ajst-9878	48	1	many	many	ADJ
ajst-9878	48	2	problems	problem	NOUN
ajst-9878	48	3	are	be	AUX
ajst-9878	48	4	based	base	VERB
ajst-9878	48	5	on	on	ADP
ajst-9878	48	6	inequalities	inequality	NOUN
ajst-9878	48	7	and	and	CCONJ
ajst-9878	48	8	are	be	AUX
ajst-9878	48	9	investigated	investigate	VERB
ajst-9878	48	10	through	through	ADP
ajst-9878	48	11	a	a	DET
ajst-9878	48	12	combination	combination	NOUN
ajst-9878	48	13	of	of	ADP
ajst-9878	48	14	different	different	ADJ
ajst-9878	48	15	knowledge	knowledge	NOUN
ajst-9878	48	16	.	.	PUNCT
ajst-9878	49	1	this	this	PRON
ajst-9878	49	2	requires	require	VERB
ajst-9878	49	3	us	we	PRON
ajst-9878	49	4	to	to	PART
ajst-9878	49	5	strengthen	strengthen	VERB
ajst-9878	49	6	the	the	DET
ajst-9878	49	7	links	link	NOUN
ajst-9878	49	8	between	between	ADP
ajst-9878	49	9	different	different	ADJ
ajst-9878	49	10	knowledge	knowledge	NOUN
ajst-9878	49	11	and	and	CCONJ
ajst-9878	49	12	abstract	abstract	ADJ
ajst-9878	49	13	real	real	ADJ
ajst-9878	49	14	-	-	PUNCT
ajst-9878	49	15	life	life	NOUN
ajst-9878	49	16	problems	problem	NOUN
ajst-9878	49	17	into	into	ADP
ajst-9878	49	18	certain	certain	ADJ
ajst-9878	49	19	basic	basic	ADJ
ajst-9878	49	20	inequality	inequality	NOUN
ajst-9878	49	21	models	model	NOUN
ajst-9878	49	22	to	to	PART
ajst-9878	49	23	improve	improve	VERB
ajst-9878	49	24	comprehensive	comprehensive	ADJ
ajst-9878	49	25	analysis	analysis	NOUN
ajst-9878	49	26	and	and	CCONJ
ajst-9878	49	27	problem	problem	NOUN
ajst-9878	49	28	-	-	PUNCT
ajst-9878	49	29	solving	solve	VERB
ajst-9878	49	30	skills	skill	NOUN
ajst-9878	49	31	.	.	PUNCT
ajst-9878	50	1	4	4	X
ajst-9878	50	2	.	.	X
ajst-9878	50	3	application	application	NOUN
ajst-9878	50	4	of	of	ADP
ajst-9878	50	5	inequalities	inequality	NOUN
ajst-9878	50	6	in	in	ADP
ajst-9878	50	7	mathematical	mathematical	ADJ
ajst-9878	50	8	problem	problem	NOUN
ajst-9878	50	9	solving	solve	VERB
ajst-9878	50	10	4.1	4.1	NUM
ajst-9878	50	11	.	.	PUNCT
ajst-9878	50	12	used	use	VERB
ajst-9878	50	13	to	to	PART
ajst-9878	50	14	find	find	VERB
ajst-9878	50	15	the	the	DET
ajst-9878	50	16	maximum	maximum	ADJ
ajst-9878	50	17	value	value	NOUN
ajst-9878	50	18	in	in	ADP
ajst-9878	50	19	particular	particular	ADJ
ajst-9878	50	20	,	,	PUNCT
ajst-9878	50	21	when	when	SCONJ
ajst-9878	50	22	it	it	PRON
ajst-9878	50	23	comes	come	VERB
ajst-9878	50	24	to	to	ADP
ajst-9878	50	25	finding	find	VERB
ajst-9878	50	26	the	the	DET
ajst-9878	50	27	maximum	maximum	ADJ
ajst-9878	50	28	value	value	NOUN
ajst-9878	50	29	of	of	ADP
ajst-9878	50	30	a	a	DET
ajst-9878	50	31	single	single	ADJ
ajst-9878	50	32	parameter	parameter	NOUN
ajst-9878	50	33	,	,	PUNCT
ajst-9878	50	34	it	it	PRON
ajst-9878	50	35	is	be	AUX
ajst-9878	50	36	common	common	ADJ
ajst-9878	50	37	to	to	PART
ajst-9878	50	38	separate	separate	VERB
ajst-9878	50	39	the	the	DET
ajst-9878	50	40	parameters	parameter	NOUN
ajst-9878	50	41	and	and	CCONJ
ajst-9878	50	42	then	then	ADV
ajst-9878	50	43	combine	combine	VERB
ajst-9878	50	44	them	they	PRON
ajst-9878	50	45	with	with	ADP
ajst-9878	50	46	known	know	VERB
ajst-9878	50	47	conditions	condition	NOUN
ajst-9878	50	48	to	to	PART
ajst-9878	50	49	create	create	VERB
ajst-9878	50	50	a	a	DET
ajst-9878	50	51	basic	basic	ADJ
ajst-9878	50	52	inequality	inequality	NOUN
ajst-9878	50	53	,	,	PUNCT
ajst-9878	50	54	which	which	PRON
ajst-9878	50	55	can	can	AUX
ajst-9878	50	56	be	be	AUX
ajst-9878	50	57	solved	solve	VERB
ajst-9878	50	58	using	use	VERB
ajst-9878	50	59	basic	basic	ADJ
ajst-9878	50	60	inequalities	inequality	NOUN
ajst-9878	50	61	.	.	PUNCT
ajst-9878	51	1	the	the	DET
ajst-9878	51	2	use	use	NOUN
ajst-9878	51	3	of	of	ADP
ajst-9878	51	4	basic	basic	ADJ
ajst-9878	51	5	inequalities	inequality	NOUN
ajst-9878	51	6	to	to	PART
ajst-9878	51	7	find	find	VERB
ajst-9878	51	8	the	the	DET
ajst-9878	51	9	most	most	ADJ
ajst-9878	51	10	value	value	NOUN
ajst-9878	51	11	should	should	AUX
ajst-9878	51	12	ensure	ensure	VERB
ajst-9878	51	13	that	that	SCONJ
ajst-9878	51	14	it	it	PRON
ajst-9878	51	15	meets	meet	VERB
ajst-9878	51	16	the	the	DET
ajst-9878	51	17	"	"	PUNCT
ajst-9878	51	18	one	one	NUM
ajst-9878	51	19	positive	positive	ADJ
ajst-9878	51	20	,	,	PUNCT
ajst-9878	51	21	two	two	NUM
ajst-9878	51	22	definite	definite	ADJ
ajst-9878	51	23	,	,	PUNCT
ajst-9878	51	24	three	three	NUM
ajst-9878	51	25	equal	equal	ADJ
ajst-9878	51	26	"	"	PUNCT
ajst-9878	51	27	,	,	PUNCT
ajst-9878	51	28	especially	especially	ADV
ajst-9878	51	29	not	not	PART
ajst-9878	51	30	to	to	PART
ajst-9878	51	31	be	be	AUX
ajst-9878	51	32	rigidly	rigidly	ADV
ajst-9878	51	33	applied	apply	VERB
ajst-9878	51	34	,	,	PUNCT
ajst-9878	51	35	otherwise	otherwise	ADV
ajst-9878	51	36	you	you	PRON
ajst-9878	51	37	may	may	AUX
ajst-9878	51	38	get	get	VERB
ajst-9878	51	39	the	the	DET
ajst-9878	51	40	wrong	wrong	NOUN
ajst-9878	51	41	.	.	PUNCT
ajst-9878	52	1	example	example	NOUN
ajst-9878	53	1	1	1	NUM
ajst-9878	53	2	knowing	know	VERB
ajst-9878	53	3	that	that	SCONJ
ajst-9878	53	4	x	x	PROPN
ajst-9878	53	5	,	,	PUNCT
ajst-9878	53	6	y	y	PROPN
ajst-9878	53	7	are	be	AUX
ajst-9878	53	8	positive	positive	ADJ
ajst-9878	53	9	real	real	ADJ
ajst-9878	53	10	numbers	number	NOUN
ajst-9878	53	11	and	and	CCONJ
ajst-9878	53	12	satisfy	satisfy	VERB
ajst-9878	53	13	the	the	DET
ajst-9878	53	14	relation	relation	NOUN
ajst-9878	53	15	x	x	PUNCT
ajst-9878	54	1	+	+	CCONJ
ajst-9878	54	2	y	y	PROPN
ajst-9878	54	3	+	+	CCONJ
ajst-9878	54	4	3	3	NUM
ajst-9878	54	5	=	=	SYM
ajst-9878	54	6	xy	xy	PROPN
ajst-9878	54	7	,	,	PUNCT
ajst-9878	54	8	and	and	CCONJ
ajst-9878	54	9	that	that	SCONJ
ajst-9878	54	10	(	(	PUNCT
ajst-9878	54	11	x	x	X
ajst-9878	54	12	+	+	PUNCT
ajst-9878	54	13	y)2	y)2	NOUN
ajst-9878	54	14	-	-	PUNCT
ajst-9878	54	15	a(x	a(x	NOUN
ajst-9878	54	16	+	+	CCONJ
ajst-9878	54	17	y	y	NOUN
ajst-9878	54	18	)	)	PUNCT
ajst-9878	54	19	+	+	CCONJ
ajst-9878	54	20	6	6	NUM
ajst-9878	54	21	≥	≥	NOUN
ajst-9878	54	22	0	0	NUM
ajst-9878	54	23	is	be	AUX
ajst-9878	54	24	constant	constant	ADJ
ajst-9878	54	25	for	for	SCONJ
ajst-9878	54	26	any	any	DET
ajst-9878	54	27	x	x	NOUN
ajst-9878	54	28	,	,	PUNCT
ajst-9878	54	29	y	y	PROPN
ajst-9878	54	30	,	,	PUNCT
ajst-9878	54	31	the	the	DET
ajst-9878	54	32	maximum	maximum	ADJ
ajst-9878	54	33	value	value	NOUN
ajst-9878	54	34	of	of	ADP
ajst-9878	54	35	the	the	DET
ajst-9878	54	36	real	real	ADJ
ajst-9878	54	37	number	number	NOUN
ajst-9878	54	38	a	a	PRON
ajst-9878	54	39	is	be	AUX
ajst-9878	54	40	_	_	NOUN
ajst-9878	55	1	_	_	PUNCT
ajst-9878	56	1	_	_	PUNCT
ajst-9878	57	1	_	_	PUNCT
ajst-9878	57	2	.	.	PUNCT
ajst-9878	58	1	the	the	DET
ajst-9878	58	2	question	question	NOUN
ajst-9878	58	3	does	do	AUX
ajst-9878	58	4	not	not	PART
ajst-9878	58	5	satisfy	satisfy	VERB
ajst-9878	58	6	the	the	DET
ajst-9878	58	7	condition	condition	NOUN
ajst-9878	58	8	of	of	ADP
ajst-9878	58	9	using	use	VERB
ajst-9878	58	10	the	the	DET
ajst-9878	58	11	basic	basic	ADJ
ajst-9878	58	12	inequality	inequality	NOUN
ajst-9878	58	13	,	,	PUNCT
ajst-9878	58	14	at	at	ADP
ajst-9878	58	15	this	this	DET
ajst-9878	58	16	point	point	NOUN
ajst-9878	58	17	you	you	PRON
ajst-9878	58	18	need	need	VERB
ajst-9878	58	19	to	to	PART
ajst-9878	58	20	use	use	VERB
ajst-9878	58	21	the	the	DET
ajst-9878	58	22	function	function	NOUN
ajst-9878	58	23	properties	property	NOUN
ajst-9878	58	24	to	to	PART
ajst-9878	58	25	break	break	VERB
ajst-9878	58	26	through	through	ADP
ajst-9878	58	27	.	.	PUNCT
ajst-9878	59	1	analysis	analysis	NOUN
ajst-9878	59	2	since	since	SCONJ
ajst-9878	59	3	the	the	DET
ajst-9878	59	4	positive	positive	ADJ
ajst-9878	59	5	real	real	ADJ
ajst-9878	59	6	numbers	number	NOUN
ajst-9878	59	7	x	x	X
ajst-9878	59	8	,	,	PUNCT
ajst-9878	59	9	y	y	PROPN
ajst-9878	59	10	satisfy	satisfy	VERB
ajst-9878	59	11	the	the	DET
ajst-9878	59	12	relation	relation	NOUN
ajst-9878	59	13	x+y+3	x+y+3	NOUN
ajst-9878	60	1	=	=	SYM
ajst-9878	60	2	xy	xy	PROPN
ajst-9878	60	3	,	,	PUNCT
ajst-9878	60	4	xy≤	xy≤	PROPN
ajst-9878	60	5	（	（	X
ajst-9878	61	1	𝐱	𝐱	PROPN
ajst-9878	61	2	𝐲	𝐲	ADP
ajst-9878	61	3	𝟐	𝟐	NUM
ajst-9878	61	4	）	）	SYM
ajst-9878	61	5	𝟐	𝟐	NUM
ajst-9878	61	6	,	,	PUNCT
ajst-9878	61	7	so	so	SCONJ
ajst-9878	61	8	x+y+3≤	x+y+3≤	PROPN
ajst-9878	61	9	（	（	PROPN
ajst-9878	61	10	𝐱	𝐱	PROPN
ajst-9878	61	11	𝐲	𝐲	ADP
ajst-9878	61	12	𝟐	𝟐	NUM
ajst-9878	61	13	）	）	SYM
ajst-9878	61	14	𝟐	𝟐	NUM
ajst-9878	61	15	that	that	ADV
ajst-9878	61	16	is	be	AUX
ajst-9878	61	17	,	,	PUNCT
ajst-9878	61	18	-4	-4	PUNCT
ajst-9878	61	19	(	(	PUNCT
ajst-9878	61	20	x+y	x+y	NUM
ajst-9878	61	21	)	)	PUNCT
ajst-9878	61	22	-12≥0	-12≥0	PROPN
ajst-9878	61	23	.	.	PUNCT
ajst-9878	62	1	solve	solve	VERB
ajst-9878	62	2	for	for	ADP
ajst-9878	62	3	x+y≥6	x+y≥6	NOUN
ajst-9878	62	4	or	or	CCONJ
ajst-9878	62	5	x+y≤-2	x+y≤-2	NOUN
ajst-9878	62	6	(	(	PUNCT
ajst-9878	62	7	round	round	ADV
ajst-9878	62	8	off	off	ADV
ajst-9878	62	9	)	)	PUNCT
ajst-9878	62	10	.	.	PUNCT
ajst-9878	63	1	and	and	CCONJ
ajst-9878	63	2	because	because	SCONJ
ajst-9878	63	3	（	（	PROPN
ajst-9878	63	4	𝐱	𝐱	PROPN
ajst-9878	63	5	𝐲	𝐲	PROPN
ajst-9878	63	6	）	）	PROPN
ajst-9878	63	7	𝟐	𝟐	NUM
ajst-9878	63	8	-a	-a	PUNCT
ajst-9878	63	9	(	(	PUNCT
ajst-9878	63	10	x+y	x+y	NUM
ajst-9878	63	11	)	)	PUNCT
ajst-9878	63	12	+6≥0	+6≥0	X
ajst-9878	63	13	is	be	AUX
ajst-9878	63	14	always	always	ADV
ajst-9878	63	15	true	true	ADJ
ajst-9878	63	16	,	,	PUNCT
ajst-9878	63	17	a≤x+y+	a≤x+y+	VERB
ajst-9878	63	18	𝟔	𝟔	NUM
ajst-9878	63	19	𝐱	𝐱	PROPN
ajst-9878	63	20	𝐲	𝐲	PROPN
ajst-9878	63	21	is	be	AUX
ajst-9878	63	22	always	always	ADV
ajst-9878	63	23	true	true	ADJ
ajst-9878	63	24	.	.	PUNCT
ajst-9878	64	1	the	the	DET
ajst-9878	64	2	problem	problem	NOUN
ajst-9878	64	3	is	be	AUX
ajst-9878	64	4	converted	convert	VERB
ajst-9878	64	5	to	to	ADP
ajst-9878	64	6	finding	find	VERB
ajst-9878	64	7	the	the	DET
ajst-9878	64	8	minimum	minimum	ADJ
ajst-9878	64	9	value	value	NOUN
ajst-9878	64	10	of	of	ADP
ajst-9878	64	11	x+y+	x+y+	PROPN
ajst-9878	64	12	𝟔	𝟔	NUM
ajst-9878	64	13	𝐱	𝐱	PROPN
ajst-9878	64	14	𝐲	𝐲	PROPN
ajst-9878	64	15	.	.	PUNCT
ajst-9878	65	1	let	let	VERB
ajst-9878	65	2	t=	t=	PRON
ajst-9878	65	3	x+y	x+y	PROPN
ajst-9878	65	4	(	(	PUNCT
ajst-9878	65	5	t≥6	t≥6	NOUN
ajst-9878	65	6	)	)	PUNCT
ajst-9878	65	7	,	,	PUNCT
ajst-9878	65	8	then	then	ADV
ajst-9878	65	9	.	.	PUNCT
ajst-9878	66	1	（	（	PUNCT
ajst-9878	66	2	𝐭	𝐭	X
ajst-9878	66	3	）	）	PROPN
ajst-9878	66	4	𝐭	𝐭	ADP
ajst-9878	66	5	𝟔	𝟔	NUM
ajst-9878	66	6	𝐭	𝐭	NOUN
ajst-9878	66	7	.	.	PUNCT
ajst-9878	67	1	from	from	ADP
ajst-9878	67	2	the	the	DET
ajst-9878	67	3	property	property	NOUN
ajst-9878	67	4	of	of	ADP
ajst-9878	67	5	the	the	DET
ajst-9878	67	6	check	check	NOUN
ajst-9878	67	7	function	function	NOUN
ajst-9878	67	8	,	,	PUNCT
ajst-9878	67	9	it	it	PRON
ajst-9878	67	10	can	can	AUX
ajst-9878	67	11	be	be	AUX
ajst-9878	67	12	seen	see	VERB
ajst-9878	67	13	that	that	SCONJ
ajst-9878	67	14	it	it	PRON
ajst-9878	67	15	is	be	AUX
ajst-9878	67	16	monotonically	monotonically	ADV
ajst-9878	67	17	increasing	increase	VERB
ajst-9878	67	18	on	on	ADP
ajst-9878	67	19	[	[	X
ajst-9878	67	20	6	6	NUM
ajst-9878	67	21	,	,	PUNCT
ajst-9878	67	22	+	+	NOUN
ajst-9878	67	23	∞	∞	NOUN
ajst-9878	67	24	)	)	PUNCT
ajst-9878	67	25	,	,	PUNCT
ajst-9878	67	26	then	then	ADV
ajst-9878	67	27	𝑓(t)min	𝑓(t)min	PROPN
ajst-9878	67	28	=	=	NOUN
ajst-9878	67	29	7	7	X
ajst-9878	67	30	.	.	PUNCT
ajst-9878	68	1	therefore	therefore	ADV
ajst-9878	68	2	,	,	PUNCT
ajst-9878	68	3	the	the	DET
ajst-9878	68	4	maximum	maximum	ADJ
ajst-9878	68	5	value	value	NOUN
ajst-9878	68	6	of	of	ADP
ajst-9878	68	7	a	a	PRON
ajst-9878	68	8	is	be	AUX
ajst-9878	68	9	7	7	NUM
ajst-9878	68	10	.	.	ADP
ajst-9878	68	11	4.2	4.2	NUM
ajst-9878	68	12	.	.	PUNCT
ajst-9878	69	1	used	use	VERB
ajst-9878	69	2	to	to	PART
ajst-9878	69	3	compare	compare	VERB
ajst-9878	69	4	sizes	size	NOUN
ajst-9878	69	5	there	there	PRON
ajst-9878	69	6	are	be	VERB
ajst-9878	69	7	two	two	NUM
ajst-9878	69	8	main	main	ADJ
ajst-9878	69	9	ways	way	NOUN
ajst-9878	69	10	to	to	PART
ajst-9878	69	11	compare	compare	VERB
ajst-9878	69	12	sizes	size	NOUN
ajst-9878	69	13	using	use	VERB
ajst-9878	69	14	inequalities	inequality	NOUN
ajst-9878	69	15	:	:	PUNCT
ajst-9878	69	16	train	train	NOUN
ajst-9878	69	17	of	of	ADP
ajst-9878	69	18	thought	thought	NOUN
ajst-9878	69	19	1	1	NUM
ajst-9878	69	20	,	,	PUNCT
ajst-9878	69	21	using	use	VERB
ajst-9878	69	22	the	the	DET
ajst-9878	69	23	monotonicity	monotonicity	NOUN
ajst-9878	69	24	of	of	ADP
ajst-9878	69	25	the	the	DET
ajst-9878	69	26	learned	learn	VERB
ajst-9878	69	27	function	function	NOUN
ajst-9878	69	28	to	to	PART
ajst-9878	69	29	make	make	VERB
ajst-9878	69	30	a	a	DET
ajst-9878	69	31	direct	direct	ADJ
ajst-9878	69	32	comparison	comparison	NOUN
ajst-9878	69	33	.	.	PUNCT
ajst-9878	70	1	train	train	NOUN
ajst-9878	70	2	of	of	ADP
ajst-9878	70	3	thought	thought	NOUN
ajst-9878	70	4	2	2	NUM
ajst-9878	70	5	,	,	PUNCT
ajst-9878	70	6	construct	construct	VERB
ajst-9878	70	7	a	a	DET
ajst-9878	70	8	new	new	ADJ
ajst-9878	70	9	function	function	NOUN
ajst-9878	70	10	,	,	PUNCT
ajst-9878	70	11	use	use	VERB
ajst-9878	70	12	the	the	DET
ajst-9878	70	13	derivative	derivative	NOUN
ajst-9878	70	14	to	to	PART
ajst-9878	70	15	study	study	VERB
ajst-9878	70	16	its	its	PRON
ajst-9878	70	17	monotonicity	monotonicity	NOUN
ajst-9878	70	18	in	in	ADP
ajst-9878	70	19	the	the	DET
ajst-9878	70	20	corresponding	corresponding	ADJ
ajst-9878	70	21	interval	interval	NOUN
ajst-9878	70	22	,	,	PUNCT
ajst-9878	70	23	and	and	CCONJ
ajst-9878	70	24	then	then	ADV
ajst-9878	70	25	compare	compare	VERB
ajst-9878	70	26	the	the	DET
ajst-9878	70	27	size	size	NOUN
ajst-9878	70	28	.	.	PUNCT
ajst-9878	70	29	example	example	NOUN
ajst-9878	70	30	2	2	NUM
ajst-9878	70	31	let	let	VERB
ajst-9878	70	32	a=3	a=3	PROPN
ajst-9878	70	33	,	,	PUNCT
ajst-9878	70	34	b=𝜋	b=𝜋	PROPN
ajst-9878	70	35	,	,	PUNCT
ajst-9878	70	36	c=3	c=3	PROPN
ajst-9878	70	37	,	,	PUNCT
ajst-9878	70	38	then	then	ADV
ajst-9878	70	39	the	the	DET
ajst-9878	70	40	magnitude	magnitude	NOUN
ajst-9878	70	41	relation	relation	NOUN
ajst-9878	70	42	of	of	ADP
ajst-9878	70	43	a	a	DET
ajst-9878	70	44	,	,	PUNCT
ajst-9878	70	45	b	b	NOUN
ajst-9878	70	46	,	,	PUNCT
ajst-9878	70	47	c	c	PROPN
ajst-9878	70	48	is	be	AUX
ajst-9878	70	49	(	(	PUNCT
ajst-9878	70	50	)	)	PUNCT
ajst-9878	70	51	.	.	PUNCT
ajst-9878	71	1	a.b	a.b	PROPN
ajst-9878	71	2	>	>	X
ajst-9878	71	3	a	a	X
ajst-9878	71	4	>	>	X
ajst-9878	71	5	c	c	PROPN
ajst-9878	71	6	b.c	b.c	PROPN
ajst-9878	71	7	>	>	X
ajst-9878	71	8	a	a	PROPN
ajst-9878	71	9	>	>	X
ajst-9878	71	10	b	b	X
ajst-9878	71	11	c.a	c.a	PROPN
ajst-9878	71	12	>	>	X
ajst-9878	71	13	b	b	PROPN
ajst-9878	71	14	>	>	X
ajst-9878	71	15	c	c	PROPN
ajst-9878	71	16	d.b	d.b	X
ajst-9878	71	17	>	>	X
ajst-9878	71	18	c	c	X
ajst-9878	71	19	>	>	X
ajst-9878	71	20	a	a	DET
ajst-9878	71	21	observation	observation	NOUN
ajst-9878	71	22	shows	show	VERB
ajst-9878	71	23	that	that	SCONJ
ajst-9878	71	24	a	a	PRON
ajst-9878	71	25	and	and	CCONJ
ajst-9878	71	26	c	c	NOUN
ajst-9878	71	27	,	,	PUNCT
ajst-9878	71	28	b	b	PROPN
ajst-9878	71	29	and	and	CCONJ
ajst-9878	71	30	c	c	PROPN
ajst-9878	71	31	can	can	AUX
ajst-9878	71	32	be	be	AUX
ajst-9878	71	33	directly	directly	ADV
ajst-9878	71	34	compared	compare	VERB
ajst-9878	71	35	by	by	ADP
ajst-9878	71	36	using	use	VERB
ajst-9878	71	37	the	the	DET
ajst-9878	71	38	exponential	exponential	ADJ
ajst-9878	71	39	function	function	NOUN
ajst-9878	71	40	power	power	NOUN
ajst-9878	71	41	function	function	NOUN
ajst-9878	71	42	property	property	NOUN
ajst-9878	71	43	.	.	PUNCT
ajst-9878	72	1	to	to	PART
ajst-9878	72	2	compare	compare	VERB
ajst-9878	72	3	the	the	DET
ajst-9878	72	4	size	size	NOUN
ajst-9878	72	5	of	of	ADP
ajst-9878	72	6	a	a	PRON
ajst-9878	72	7	and	and	CCONJ
ajst-9878	72	8	b	b	NOUN
ajst-9878	72	9	,	,	PUNCT
ajst-9878	72	10	we	we	PRON
ajst-9878	72	11	need	need	VERB
ajst-9878	72	12	to	to	PART
ajst-9878	72	13	construct	construct	VERB
ajst-9878	72	14	the	the	DET
ajst-9878	72	15	function	function	NOUN
ajst-9878	72	16	and	and	CCONJ
ajst-9878	72	17	use	use	VERB
ajst-9878	72	18	the	the	DET
ajst-9878	72	19	derivative	derivative	ADJ
ajst-9878	72	20	knowledge	knowledge	NOUN
ajst-9878	72	21	analysis	analysis	NOUN
ajst-9878	72	22	.	.	PUNCT
ajst-9878	73	1	analysis	analysis	NOUN
ajst-9878	73	2	the	the	DET
ajst-9878	73	3	power	power	NOUN
ajst-9878	73	4	function	function	NOUN
ajst-9878	73	5	y=	y=	NUM
ajst-9878	73	6	𝒙𝟑	𝒙𝟑	PROPN
ajst-9878	73	7	increases	increase	VERB
ajst-9878	73	8	monotonically	monotonically	ADV
ajst-9878	73	9	on	on	ADP
ajst-9878	73	10	(	(	PUNCT
ajst-9878	73	11	0	0	NUM
ajst-9878	73	12	,	,	PUNCT
ajst-9878	73	13	+	+	ADJ
ajst-9878	73	14	)	)	PUNCT
ajst-9878	73	15	,	,	PUNCT
ajst-9878	73	16	and	and	CCONJ
ajst-9878	73	17	π	π	X
ajst-9878	73	18	>	>	X
ajst-9878	73	19	3	3	NUM
ajst-9878	73	20	,	,	PUNCT
ajst-9878	73	21	b	b	PROPN
ajst-9878	73	22	>	>	X
ajst-9878	73	23	c.	c.	PROPN
ajst-9878	73	24	and	and	CCONJ
ajst-9878	73	25	because	because	SCONJ
ajst-9878	73	26	y=𝑥	y=𝑥	PROPN
ajst-9878	73	27	increases	increase	VERB
ajst-9878	73	28	monotone	monotone	ADJ
ajst-9878	73	29	over	over	ADP
ajst-9878	73	30	r	r	NOUN
ajst-9878	73	31	,	,	PUNCT
ajst-9878	73	32	π	π	X
ajst-9878	73	33	>	>	X
ajst-9878	73	34	3	3	NUM
ajst-9878	73	35	,	,	PUNCT
ajst-9878	73	36	so	so	ADV
ajst-9878	73	37	a	a	DET
ajst-9878	73	38	>	>	X
ajst-9878	73	39	c.	c.	NOUN
ajst-9878	73	40	since	since	SCONJ
ajst-9878	73	41	a=3	a=3	PROPN
ajst-9878	73	42	,	,	PUNCT
ajst-9878	73	43	b=𝜋	b=𝜋	PROPN
ajst-9878	73	44	,	,	PUNCT
ajst-9878	73	45	take	take	VERB
ajst-9878	73	46	the	the	DET
ajst-9878	73	47	logarithm	logarithm	NOUN
ajst-9878	73	48	of	of	ADP
ajst-9878	73	49	both	both	DET
ajst-9878	73	50	sides	side	NOUN
ajst-9878	73	51	of	of	ADP
ajst-9878	73	52	the	the	DET
ajst-9878	73	53	equation	equation	NOUN
ajst-9878	73	54	to	to	PART
ajst-9878	73	55	get	get	VERB
ajst-9878	73	56	lna	lna	PROPN
ajst-9878	73	57	=	=	NOUN
ajst-9878	73	58	πln3	πln3	PROPN
ajst-9878	73	59	,	,	PUNCT
ajst-9878	73	60	lnb=3lnπ	lnb=3lnπ	X
ajst-9878	73	61	.	.	PUNCT
ajst-9878	74	1	let	let	VERB
ajst-9878	75	1	f	f	PROPN
ajst-9878	75	2	（	（	PROPN
ajst-9878	75	3	t	t	PROPN
ajst-9878	75	4	）	）	PUNCT
ajst-9878	76	1	=	=	PRON
ajst-9878	76	2	,	,	PUNCT
ajst-9878	76	3	then	then	ADV
ajst-9878	76	4	,	,	PUNCT
ajst-9878	76	5	𝑥	𝑥	X
ajst-9878	76	6	/	/	SYM
ajst-9878	76	7	,	,	PUNCT
ajst-9878	76	8	then	then	ADV
ajst-9878	76	9	f	f	X
ajst-9878	76	10	(	(	PUNCT
ajst-9878	76	11	x	x	X
ajst-9878	76	12	)	)	PUNCT
ajst-9878	76	13	is	be	AUX
ajst-9878	76	14	monotonically	monotonically	ADV
ajst-9878	76	15	decreasing	decrease	VERB
ajst-9878	76	16	.	.	PUNCT
ajst-9878	77	1	i.e.	i.e.	X
ajst-9878	77	2	f	f	X
ajst-9878	77	3	(	(	PUNCT
ajst-9878	77	4	3	3	NUM
ajst-9878	77	5	)	)	PUNCT
ajst-9878	77	6	>	>	X
ajst-9878	77	7	f	f	PROPN
ajst-9878	77	8	(	(	PUNCT
ajst-9878	77	9	π	π	PROPN
ajst-9878	77	10	)	)	PUNCT
ajst-9878	77	11	,	,	PUNCT
ajst-9878	77	12	i.e.	i.e.	X
ajst-9878	77	13	＞	＞	X
ajst-9878	77	14	,	,	PUNCT
ajst-9878	77	15	i.e.	i.e.	X
ajst-9878	77	16	πln3	πln3	X
ajst-9878	77	17	>	>	X
ajst-9878	77	18	3lnπ	3lnπ	PROPN
ajst-9878	77	19	,	,	PUNCT
ajst-9878	77	20	lna	lna	PROPN
ajst-9878	77	21	>	>	X
ajst-9878	77	22	lnb	lnb	PROPN
ajst-9878	77	23	that	that	PRON
ajst-9878	77	24	is	be	AUX
ajst-9878	77	25	,	,	PUNCT
ajst-9878	77	26	a	a	DET
ajst-9878	77	27	>	>	X
ajst-9878	77	28	b	b	PROPN
ajst-9878	77	29	,	,	PUNCT
ajst-9878	77	30	so	so	ADV
ajst-9878	77	31	a	a	DET
ajst-9878	77	32	>	>	X
ajst-9878	77	33	b	b	X
ajst-9878	77	34	>	>	X
ajst-9878	77	35	c	c	X
ajst-9878	77	36	,	,	PUNCT
ajst-9878	77	37	so	so	ADV
ajst-9878	77	38	choose	choose	VERB
ajst-9878	77	39	c.	c.	PROPN
ajst-9878	77	40	4.3	4.3	NUM
ajst-9878	77	41	.	.	PUNCT
ajst-9878	78	1	used	use	VERB
ajst-9878	78	2	to	to	PART
ajst-9878	78	3	find	find	VERB
ajst-9878	78	4	the	the	DET
ajst-9878	78	5	value	value	NOUN
ajst-9878	78	6	range	range	NOUN
ajst-9878	78	7	solving	solve	VERB
ajst-9878	78	8	the	the	DET
ajst-9878	78	9	range	range	NOUN
ajst-9878	78	10	of	of	ADP
ajst-9878	78	11	parameter	parameter	NOUN
ajst-9878	78	12	values	value	NOUN
ajst-9878	78	13	is	be	AUX
ajst-9878	78	14	an	an	DET
ajst-9878	78	15	important	important	ADJ
ajst-9878	78	16	problem	problem	NOUN
ajst-9878	78	17	in	in	ADP
ajst-9878	78	18	high	high	ADJ
ajst-9878	78	19	school	school	NOUN
ajst-9878	78	20	mathematics	mathematic	NOUN
ajst-9878	78	21	.	.	PUNCT
ajst-9878	79	1	the	the	DET
ajst-9878	79	2	situation	situation	NOUN
ajst-9878	79	3	of	of	ADP
ajst-9878	79	4	related	relate	VERB
ajst-9878	79	5	exercises	exercise	NOUN
ajst-9878	79	6	is	be	AUX
ajst-9878	79	7	complex	complex	ADJ
ajst-9878	79	8	and	and	CCONJ
ajst-9878	79	9	changeable	changeable	ADJ
ajst-9878	79	10	,	,	PUNCT
ajst-9878	79	11	and	and	CCONJ
ajst-9878	79	12	the	the	DET
ajst-9878	79	13	ideas	idea	NOUN
ajst-9878	79	14	of	of	ADP
ajst-9878	79	15	solving	solve	VERB
ajst-9878	79	16	problems	problem	NOUN
ajst-9878	79	17	are	be	AUX
ajst-9878	79	18	flexible	flexible	ADJ
ajst-9878	79	19	and	and	CCONJ
ajst-9878	79	20	diverse	diverse	ADJ
ajst-9878	79	21	,	,	PUNCT
ajst-9878	79	22	which	which	PRON
ajst-9878	79	23	can	can	AUX
ajst-9878	79	24	sometimes	sometimes	ADV
ajst-9878	79	25	get	get	VERB
ajst-9878	79	26	unexpected	unexpected	ADJ
ajst-9878	79	27	results	result	NOUN
ajst-9878	79	28	by	by	ADP
ajst-9878	79	29	cutting	cut	VERB
ajst-9878	79	30	into	into	ADP
ajst-9878	79	31	the	the	DET
ajst-9878	79	32	perspective	perspective	NOUN
ajst-9878	79	33	of	of	ADP
ajst-9878	79	34	nonequality	nonequality	NOUN
ajst-9878	79	35	.	.	PUNCT
ajst-9878	80	1	example	example	NOUN
ajst-9878	80	2	3	3	NUM
ajst-9878	80	3	given	give	VERB
ajst-9878	80	4	x	x	PUNCT
ajst-9878	80	5	>	>	X
ajst-9878	80	6	0	0	PROPN
ajst-9878	80	7	,	,	PUNCT
ajst-9878	80	8	y	y	PROPN
ajst-9878	80	9	>	>	X
ajst-9878	80	10	0	0	PROPN
ajst-9878	80	11	,	,	PUNCT
ajst-9878	80	12	x+y+xy=4	x+y+xy=4	PROPN
ajst-9878	80	13	,	,	PUNCT
ajst-9878	80	14	then	then	ADV
ajst-9878	80	15	the	the	DET
ajst-9878	80	16	range	range	NOUN
ajst-9878	80	17	of	of	ADP
ajst-9878	80	18	𝐱𝐲	𝐱𝐲	X
ajst-9878	80	19	𝟏	𝟏	NUM
ajst-9878	80	20	𝐱𝟐𝐲𝟐	𝐱𝟐𝐲𝟐	NOUN
ajst-9878	80	21	𝟐𝐱𝐲	𝟐𝐱𝐲	NOUN
ajst-9878	80	22	𝟏𝟕	𝟏𝟕	NUM
ajst-9878	80	23	value	value	NOUN
ajst-9878	80	24	is	be	AUX
ajst-9878	80	25	.	.	PUNCT
ajst-9878	81	1	this	this	DET
ajst-9878	81	2	problem	problem	NOUN
ajst-9878	81	3	has	have	VERB
ajst-9878	81	4	a	a	DET
ajst-9878	81	5	certain	certain	ADJ
ajst-9878	81	6	skill	skill	NOUN
ajst-9878	81	7	,	,	PUNCT
ajst-9878	81	8	it	it	PRON
ajst-9878	81	9	needs	need	VERB
ajst-9878	81	10	to	to	PART
ajst-9878	81	11	use	use	VERB
ajst-9878	81	12	the	the	DET
ajst-9878	81	13	knowledge	knowledge	NOUN
ajst-9878	81	14	of	of	ADP
ajst-9878	81	15	inequality	inequality	NOUN
ajst-9878	81	16	to	to	PART
ajst-9878	81	17	construct	construct	VERB
ajst-9878	81	18	a	a	DET
ajst-9878	81	19	new	new	ADJ
ajst-9878	81	20	inequality	inequality	NOUN
ajst-9878	81	21	relation	relation	NOUN
ajst-9878	81	22	,	,	PUNCT
ajst-9878	81	23	and	and	CCONJ
ajst-9878	81	24	then	then	ADV
ajst-9878	81	25	combine	combine	VERB
ajst-9878	81	26	the	the	DET
ajst-9878	81	27	properties	property	NOUN
ajst-9878	81	28	of	of	ADP
ajst-9878	81	29	function	function	NOUN
ajst-9878	81	30	to	to	PART
ajst-9878	81	31	find	find	VERB
ajst-9878	81	32	its	its	PRON
ajst-9878	81	33	maximum	maximum	ADJ
ajst-9878	81	34	value	value	NOUN
ajst-9878	81	35	.	.	PUNCT
ajst-9878	82	1	analysis	analysis	NOUN
ajst-9878	82	2	because	because	SCONJ
ajst-9878	82	3	x	x	PROPN
ajst-9878	82	4	>	>	X
ajst-9878	82	5	0	0	PROPN
ajst-9878	82	6	,	,	PUNCT
ajst-9878	82	7	y	y	PROPN
ajst-9878	82	8	>	>	X
ajst-9878	82	9	0	0	PROPN
ajst-9878	82	10	,	,	PUNCT
ajst-9878	82	11	x+y+xy=4	x+y+xy=4	PROPN
ajst-9878	82	12	,	,	PUNCT
ajst-9878	82	13	x+	x+	X
ajst-9878	82	14	𝟏	𝟏	PROPN
ajst-9878	82	15	𝟐	𝟐	NUM
ajst-9878	82	16	y+xy≥2	y+xy≥2	NOUN
ajst-9878	82	17	𝟏	𝟏	NUM
ajst-9878	82	18	𝟐	𝟐	NUM
ajst-9878	82	19	𝐱𝐲+xy=√𝟐	𝐱𝐲+xy=√𝟐	NOUN
ajst-9878	82	20	*	*	NOUN
ajst-9878	82	21	𝐱𝐲+xy	𝐱𝐲+xy	NOUN
ajst-9878	82	22	.	.	PUNCT
ajst-9878	83	1	that	that	PRON
ajst-9878	83	2	is	be	AUX
ajst-9878	83	3	,	,	PUNCT
ajst-9878	83	4	（	（	PUNCT
ajst-9878	84	1	xy	xy	PROPN
ajst-9878	84	2	）	）	PROPN
ajst-9878	85	1	√2	√2	PROPN
ajst-9878	85	2	*	*	PROPN
ajst-9878	85	3	xy-4≤0	xy-4≤0	PROPN
ajst-9878	85	4	.	.	PUNCT
ajst-9878	86	1	solve	solve	VERB
ajst-9878	86	2	for	for	ADP
ajst-9878	86	3	0＜	0＜	NUM
ajst-9878	86	4	xy≤√2,that	xy≤√2,that	PROPN
ajst-9878	86	5	is,1＜xy+1≤3	is,1＜xy+1≤3	VERB
ajst-9878	86	6	.	.	PROPN
ajst-9878	87	1	166	166	NUM
ajst-9878	87	2	therefore	therefore	ADV
ajst-9878	87	3	𝐱𝐲	𝐱𝐲	NOUN
ajst-9878	87	4	𝟏	𝟏	NUM
ajst-9878	87	5	𝐱𝟐𝐲𝟐	𝐱𝟐𝐲𝟐	NOUN
ajst-9878	87	6	𝟐𝐱𝐲	𝟐𝐱𝐲	NOUN
ajst-9878	87	7	𝟏𝟕	𝟏𝟕	PROPN
ajst-9878	88	1	=	=	SYM
ajst-9878	88	2	𝟏	𝟏	PROPN
ajst-9878	88	3	𝐱𝐲	𝐱𝐲	NOUN
ajst-9878	88	4	𝟏	𝟏	NUM
ajst-9878	88	5	𝟏𝟔	𝟏𝟔	NUM
ajst-9878	88	6	𝐱𝐲	𝐱𝐲	NOUN
ajst-9878	88	7	𝟏	𝟏	PROPN
ajst-9878	88	8	.	.	PUNCT
ajst-9878	89	1	let	let	VERB
ajst-9878	89	2	t	t	NOUN
ajst-9878	89	3	=	=	SYM
ajst-9878	89	4	xy+1	xy+1	PROPN
ajst-9878	89	5	,	,	PUNCT
ajst-9878	89	6	then	then	ADV
ajst-9878	89	7	f（t）=t+	f（t）=t+	VERB
ajst-9878	89	8	𝟏𝟔	𝟏𝟔	NUM
ajst-9878	89	9	𝐭	𝐭	NOUN
ajst-9878	89	10	.	.	PUNCT
ajst-9878	90	1	from	from	ADP
ajst-9878	90	2	the	the	DET
ajst-9878	90	3	property	property	NOUN
ajst-9878	90	4	of	of	ADP
ajst-9878	90	5	the	the	DET
ajst-9878	90	6	check	check	NOUN
ajst-9878	90	7	function	function	NOUN
ajst-9878	90	8	,	,	PUNCT
ajst-9878	90	9	we	we	PRON
ajst-9878	90	10	know	know	VERB
ajst-9878	90	11	that	that	SCONJ
ajst-9878	90	12	it	it	PRON
ajst-9878	90	13	is	be	AUX
ajst-9878	90	14	monotonically	monotonically	ADV
ajst-9878	90	15	decreasing	decrease	VERB
ajst-9878	90	16	on	on	ADP
ajst-9878	90	17	(	(	PUNCT
ajst-9878	90	18	1,3	1,3	NUM
ajst-9878	90	19	]	]	PUNCT
ajst-9878	90	20	,	,	PUNCT
ajst-9878	90	21	then	then	ADV
ajst-9878	90	22	f	f	PROPN
ajst-9878	90	23	(	(	PUNCT
ajst-9878	90	24	t	t	PROPN
ajst-9878	90	25	)	)	PUNCT
ajst-9878	90	26	∈	∈	PROPN
ajst-9878	90	27	[	[	PUNCT
ajst-9878	90	28	,	,	PUNCT
ajst-9878	90	29	17	17	NUM
ajst-9878	90	30	)	)	PUNCT
ajst-9878	90	31	.	.	PUNCT
ajst-9878	91	1	then	then	ADV
ajst-9878	91	2	the	the	DET
ajst-9878	91	3	value	value	NOUN
ajst-9878	91	4	range	range	NOUN
ajst-9878	91	5	of	of	ADP
ajst-9878	91	6	the	the	DET
ajst-9878	91	7	original	original	ADJ
ajst-9878	91	8	formula	formula	NOUN
ajst-9878	91	9	is	be	AUX
ajst-9878	91	10	(	(	PUNCT
ajst-9878	91	11	,	,	PUNCT
ajst-9878	91	12	]	]	X
ajst-9878	91	13	.	.	PUNCT
ajst-9878	92	1	5	5	X
ajst-9878	92	2	.	.	X
ajst-9878	92	3	conclusion	conclusion	NOUN
ajst-9878	92	4	in	in	ADP
ajst-9878	92	5	this	this	DET
ajst-9878	92	6	paper	paper	NOUN
ajst-9878	92	7	,	,	PUNCT
ajst-9878	92	8	the	the	DET
ajst-9878	92	9	application	application	NOUN
ajst-9878	92	10	of	of	ADP
ajst-9878	92	11	inequality	inequality	NOUN
ajst-9878	92	12	and	and	CCONJ
ajst-9878	92	13	how	how	SCONJ
ajst-9878	92	14	to	to	PART
ajst-9878	92	15	master	master	VERB
ajst-9878	92	16	and	and	CCONJ
ajst-9878	92	17	use	use	VERB
ajst-9878	92	18	inequality	inequality	NOUN
ajst-9878	92	19	in	in	ADP
ajst-9878	92	20	the	the	DET
ajst-9878	92	21	ordinary	ordinary	ADJ
ajst-9878	92	22	learning	learning	NOUN
ajst-9878	92	23	process	process	NOUN
ajst-9878	92	24	are	be	AUX
ajst-9878	92	25	simply	simply	ADV
ajst-9878	92	26	discussed	discuss	VERB
ajst-9878	92	27	.	.	PUNCT
ajst-9878	93	1	by	by	ADP
ajst-9878	93	2	learning	learn	VERB
ajst-9878	93	3	the	the	DET
ajst-9878	93	4	basic	basic	ADJ
ajst-9878	93	5	knowledge	knowledge	NOUN
ajst-9878	93	6	of	of	ADP
ajst-9878	93	7	inequalities	inequality	NOUN
ajst-9878	93	8	and	and	CCONJ
ajst-9878	93	9	training	train	VERB
ajst-9878	93	10	basic	basic	ADJ
ajst-9878	93	11	skills	skill	NOUN
ajst-9878	93	12	,	,	PUNCT
ajst-9878	93	13	we	we	PRON
ajst-9878	93	14	can	can	AUX
ajst-9878	93	15	help	help	VERB
ajst-9878	93	16	improve	improve	VERB
ajst-9878	93	17	our	our	PRON
ajst-9878	93	18	logical	logical	ADJ
ajst-9878	93	19	thinking	thinking	NOUN
ajst-9878	93	20	ability	ability	NOUN
ajst-9878	93	21	and	and	CCONJ
ajst-9878	93	22	the	the	DET
ajst-9878	93	23	ability	ability	NOUN
ajst-9878	93	24	to	to	PART
ajst-9878	93	25	analyze	analyze	VERB
ajst-9878	93	26	and	and	CCONJ
ajst-9878	93	27	solve	solve	VERB
ajst-9878	93	28	practical	practical	ADJ
ajst-9878	93	29	problems	problem	NOUN
ajst-9878	93	30	,	,	PUNCT
ajst-9878	93	31	hoping	hope	VERB
ajst-9878	93	32	to	to	PART
ajst-9878	93	33	provide	provide	VERB
ajst-9878	93	34	some	some	DET
ajst-9878	93	35	help	help	NOUN
ajst-9878	93	36	for	for	ADP
ajst-9878	93	37	cultivating	cultivate	VERB
ajst-9878	93	38	and	and	CCONJ
ajst-9878	93	39	improving	improve	VERB
ajst-9878	93	40	our	our	PRON
ajst-9878	93	41	comprehensive	comprehensive	ADJ
ajst-9878	93	42	ability	ability	NOUN
ajst-9878	93	43	.	.	PUNCT
ajst-9878	94	1	references	reference	NOUN
ajst-9878	94	2	[	[	X
ajst-9878	94	3	1	1	NUM
ajst-9878	94	4	]	]	X
ajst-9878	94	5	nan	nan	PROPN
ajst-9878	94	6	shan	shan	PROPN
ajst-9878	94	7	.	.	PUNCT
ajst-9878	95	1	cauchy	cauchy	PROPN
ajst-9878	95	2	inequality	inequality	NOUN
ajst-9878	95	3	and	and	CCONJ
ajst-9878	95	4	ordering	order	VERB
ajst-9878	95	5	inequality	inequality	NOUN
ajst-9878	95	6	[	[	X
ajst-9878	95	7	m	m	X
ajst-9878	95	8	]	]	X
ajst-9878	95	9	.	.	PUNCT
ajst-9878	96	1	shanghai	shanghai	PROPN
ajst-9878	96	2	:	:	PUNCT
ajst-9878	96	3	shanghai	shanghai	PROPN
ajst-9878	96	4	education	education	PROPN
ajst-9878	96	5	press	press	PROPN
ajst-9878	96	6	,	,	PUNCT
ajst-9878	96	7	2012	2012	NUM
ajst-9878	96	8	.	.	PUNCT
ajst-9878	97	1	[	[	X
ajst-9878	97	2	2	2	X
ajst-9878	97	3	]	]	X
ajst-9878	97	4	peng	peng	PROPN
ajst-9878	97	5	zhifeng	zhifeng	PROPN
ajst-9878	97	6	.	.	PUNCT
ajst-9878	98	1	analysis	analysis	NOUN
ajst-9878	98	2	of	of	ADP
ajst-9878	98	3	mathematical	mathematical	ADJ
ajst-9878	98	4	thinking	thinking	NOUN
ajst-9878	98	5	in	in	ADP
ajst-9878	98	6	the	the	DET
ajst-9878	98	7	teaching	teaching	NOUN
ajst-9878	98	8	of	of	ADP
ajst-9878	98	9	mathematical	mathematical	ADJ
ajst-9878	98	10	inequality	inequality	NOUN
ajst-9878	98	11	in	in	ADP
ajst-9878	98	12	senior	senior	ADJ
ajst-9878	98	13	high	high	ADJ
ajst-9878	98	14	school	school	NOUN
ajst-9878	99	1	[	[	X
ajst-9878	99	2	j	j	X
ajst-9878	99	3	]	]	X
ajst-9878	99	4	.	.	PUNCT
ajst-9878	100	1	mathematical	mathematical	ADJ
ajst-9878	100	2	physics	physics	NOUN
ajst-9878	100	3	and	and	CCONJ
ajst-9878	100	4	chemistry	chemistry	NOUN
ajst-9878	100	5	of	of	ADP
ajst-9878	100	6	middle	middle	ADJ
ajst-9878	100	7	school	school	NOUN
ajst-9878	100	8	students	student	NOUN
ajst-9878	100	9	(	(	PUNCT
ajst-9878	100	10	graduate	graduate	NOUN
ajst-9878	100	11	edition),2015,06:22	edition),2015,06:22	PROPN
ajst-9878	100	12	.	.	PUNCT
ajst-9878	101	1	[	[	X
ajst-9878	101	2	3	3	X
ajst-9878	101	3	]	]	X
ajst-9878	102	1	[	[	X
ajst-9878	102	2	3]zhang	3]zhang	NUM
ajst-9878	102	3	ping	ping	NOUN
ajst-9878	102	4	,	,	PUNCT
ajst-9878	102	5	xie	xie	PROPN
ajst-9878	102	6	xiaoping	xiaoping	PROPN
ajst-9878	102	7	.	.	PUNCT
ajst-9878	103	1	[	[	X
ajst-9878	103	2	3	3	NUM
ajst-9878	103	3	]	]	X
ajst-9878	103	4	effective	effective	ADJ
ajst-9878	103	5	strategies	strategy	NOUN
ajst-9878	103	6	for	for	ADP
ajst-9878	103	7	teaching	teach	VERB
ajst-9878	103	8	inequality	inequality	NOUN
ajst-9878	103	9	in	in	ADP
ajst-9878	103	10	high	high	ADJ
ajst-9878	103	11	school	school	NOUN
ajst-9878	103	12	mathematics	mathematic	NOUN
ajst-9878	104	1	[	[	X
ajst-9878	104	2	j	j	X
ajst-9878	104	3	]	]	X
ajst-9878	104	4	.	.	PUNCT
ajst-9878	105	1	navigation	navigation	NOUN
ajst-9878	105	2	of	of	ADP
ajst-9878	105	3	the	the	DET
ajst-9878	105	4	arts	art	NOUN
ajst-9878	105	5	and	and	CCONJ
ajst-9878	105	6	sciences	sciences	PROPN
ajst-9878	105	7	(	(	PUNCT
ajst-9878	105	8	middle),2015,10:25	middle),2015,10:25	NOUN
ajst-9878	105	9	.	.	PUNCT
ajst-9878	106	1	[	[	X
ajst-9878	106	2	4	4	X
ajst-9878	106	3	]	]	X
ajst-9878	106	4	liang	liang	PROPN
ajst-9878	106	5	songlin	songlin	PROPN
ajst-9878	106	6	.	.	PUNCT
ajst-9878	107	1	some	some	DET
ajst-9878	107	2	suggestions	suggestion	NOUN
ajst-9878	107	3	on	on	ADP
ajst-9878	107	4	the	the	DET
ajst-9878	107	5	teaching	teaching	NOUN
ajst-9878	107	6	of	of	ADP
ajst-9878	107	7	mathematical	mathematical	ADJ
ajst-9878	107	8	inequalities	inequality	NOUN
ajst-9878	107	9	in	in	ADP
ajst-9878	107	10	high	high	ADJ
ajst-9878	107	11	school	school	NOUN
ajst-9878	108	1	[	[	X
ajst-9878	108	2	j	j	X
ajst-9878	108	3	]	]	X
ajst-9878	108	4	.	.	PUNCT
ajst-9878	109	1	new	new	ADJ
ajst-9878	109	2	curriculum	curriculum	NOUN
ajst-9878	109	3	learning	learn	VERB
ajst-9878	109	4	(	(	PUNCT
ajst-9878	109	5	basic	basic	ADJ
ajst-9878	109	6	education),2010,02:84	education),2010,02:84	NOUN
ajst-9878	109	7	.	.	PUNCT
ajst-9878	110	1	[	[	X
ajst-9878	110	2	5	5	X
ajst-9878	110	3	]	]	X
ajst-9878	110	4	qin	qin	PROPN
ajst-9878	110	5	xiuhong	xiuhong	PROPN
ajst-9878	110	6	.	.	PUNCT
ajst-9878	111	1	question	question	VERB
ajst-9878	111	2	analysis	analysis	NOUN
ajst-9878	111	3	and	and	CCONJ
ajst-9878	111	4	teaching	teach	VERB
ajst-9878	111	5	strategy	strategy	NOUN
ajst-9878	111	6	research	research	NOUN
ajst-9878	111	7	of	of	ADP
ajst-9878	111	8	senior	senior	ADJ
ajst-9878	111	9	high	high	ADJ
ajst-9878	111	10	school	school	NOUN
ajst-9878	111	11	mathematics	mathematic	NOUN
ajst-9878	111	12	inequality	inequality	NOUN
ajst-9878	111	13	test	test	NOUN
ajst-9878	112	1	[	[	X
ajst-9878	112	2	j	j	X
ajst-9878	112	3	]	]	X
ajst-9878	112	4	.	.	PUNCT
ajst-9878	113	1	mathematics	mathematic	NOUN
ajst-9878	113	2	learning	learning	NOUN
ajst-9878	113	3	and	and	CCONJ
ajst-9878	113	4	research,2021(13	research,2021(13	PROPN
ajst-9878	113	5	)	)	PUNCT
ajst-9878	113	6	:	:	PUNCT
ajst-9878	113	7	2	2	NUM
ajst-9878	113	8	-	-	SYM
ajst-9878	113	9	3	3	NUM
ajst-9878	113	10	.	.	PUNCT
ajst-9878	114	1	[	[	X
ajst-9878	114	2	6	6	NUM
ajst-9878	114	3	]	]	PUNCT
ajst-9878	114	4	liu	liu	PROPN
ajst-9878	114	5	qian	qian	PROPN
ajst-9878	114	6	.	.	PUNCT
ajst-9878	115	1	research	research	NOUN
ajst-9878	115	2	on	on	ADP
ajst-9878	115	3	the	the	DET
ajst-9878	115	4	solution	solution	NOUN
ajst-9878	115	5	of	of	ADP
ajst-9878	115	6	inequality	inequality	NOUN
ajst-9878	115	7	problem	problem	NOUN
ajst-9878	115	8	in	in	ADP
ajst-9878	115	9	high	high	ADJ
ajst-9878	115	10	school	school	NOUN
ajst-9878	115	11	mathematics	mathematic	NOUN
ajst-9878	116	1	[	[	X
ajst-9878	116	2	j	j	X
ajst-9878	116	3	]	]	X
ajst-9878	116	4	.	.	PUNCT
ajst-9878	117	1	reference	reference	NOUN
ajst-9878	117	2	for	for	ADP
ajst-9878	117	3	middle	middle	ADJ
ajst-9878	117	4	school	school	NOUN
ajst-9878	117	5	mathematics	mathematic	NOUN
ajst-9878	117	6	teaching,2021(09	teaching,2021(09	PROPN
ajst-9878	117	7	)	)	PUNCT
ajst-9878	117	8	:	:	PUNCT
ajst-9878	117	9	49	49	NUM
ajst-9878	117	10	-	-	SYM
ajst-9878	117	11	50	50	NUM
ajst-9878	117	12	.	.	PUNCT