Chapter One

the morning comes and misaki get up from her bed

and as usual with a blank expression she do her daily chores

then went to school not bothering to bid goodbye to her brother who was probably sleeping right now

(its just the usual huh?) misaki thought as she look at the crowd of students chatting and doing stuffs together

(whatever)

misaki walk to her classroom as always and duck her head down the table as she listen to her calm music then close her eyes

after a few minutes she heard the class went silent and so she knew the class was about to start so she quickly sat down properly and look at the front

"If x and a are two real numbers the distance between x and a is |x − a|. For most purposes in calculus it is better to think of an inequality like |x − 5| < 2 geometrically rather then algebraically. That is\, think “The number x is within 2 units of 5\,” rather than “The absolute value of x minus 5 is strictly less than 2.” The first formulation makes it clear that x is in the open interval (3\, 7)  then Let a be a real number. A neighborhood of a is an open interval (c\, d) in R which contains a. An open interval (a − δ\, a + δ) which is centered at a is a symmetric neighborhood (or a δ neighborhood) of a. next A deleted (or punctured) neighborhood of a point a ∈ R is an open interval around a from which a has been deleted. Thus\, for example\, the deleted δ-neighborhood about 3 would be (3 − δ\, 3 + δ) \ {3} or\, using different notation\, (3 − δ\, 3) ∪ (3\, 3 + δ). also A point a is an accumulation point of a set B ⊆ R if every deleted neighborhood of a contains at least one point of B.(For Set Operations). Let A and B be subsets of a set S. Then

(1) x ∈ A ∪ B if x ∈ A or x ∈ B (union);

(2) x ∈ A ∩ B if x ∈ A and x ∈ B (intersection);

(3) x ∈ A \ B if x ∈ A and x /∈ B (set difference); and

(4) x ∈ Ac

if x ∈ S \ A (complement).

If the set S is not specified, it is usually understood to be the set R of real numbers or, starting in Part 6, the set Rn, Euclidean n-dimensional space. so class do you understand the inequalities and absolute values ways? if so please answer these questions"

the questions

(1) The inequality |x − 2| < 6 can be expressed in the form a < x < b where a = ____ and b = ._____

(2) The inequality −15 ≤ x ≤ 7 can be expressed in the form |x − a| ≤ b where a = ____ and b = .____

(3) Find all numbers x which satisfy |x2 − 9| = |x2 − 5|.

Answer: x = ____ and x = .____

(4) In interval notation the solution set for the inequality (x + 1 / x − 2) ≤ (x + 2 / x + 3) is (−∞\,____ ) ∪ [ ____\, 2 ).

(5) Let a\, b ∈ R. Show that | |a| − |b| | ≤ |a − b|.

after the class all the students exhailed "that was tough" "i could barely answer it" "waah my brain is drained now!"

but none of that really bothers misaki and just sat properly then look outside the window when she saw a tiny snow

(winter is coming huh) then the next teacher came in but with another student then the students became silent again

the teacher gave a sign which the boy seem to have understand and stand firmly infront of everyone

"Hello! my name is Kiriga Takirami nice to meet you!" kiriga bow slightly as girls squealed over him so he  smiled in return

"so as everyone can see Takirami-san is gonna be your new classmate from today onwards now takirami-san please choose an empty seat" but the only empty seat that was left was misaki's side so kiriga went there and the class start

"umm hello i'm-" before he could even finish misaki bluntly reply to him with a "i know"

(that's kinda rude isn't it?) kiriga thought so he tried to speak again but once again rudely interrupted by misaki "just shut it" and take a note from the lesson

"takirami-san is there a problem ?" the teacher ask which embarrassed kiriga

(seem like she doesn't wanna be disturbed while studying)

"no ma'am there isn't" kiriga smiled to reassure the teacher and so the lesson starts again

(what a fake) misaki thought as she ignore him for the whole day

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