Iboralar trenajyori

dan oshadi / dan katta

Bir miqdor boshqasidan ma’lum songa ko‘p ekanini bildiradi.

“The quantity x exceeds y by 7.”

x = y + 7

⚠ y = x + 7 deb yozish — munosabat teskari bo‘lib qoladi.

dan katta

Qat’iy tengsizlik: teng bo‘lish mumkin emas.

“The value of 3x is greater than 12.”

3x > 12

⚠ ≥ belgisini ishlatish. “Greater than” tenglikni o‘z ichiga olmaydi.

dan kichik

Qat’iy tengsizlik: chap tomon o‘ng tomondan kichik.

“The number n is less than 9.”

n < 9

⚠ “less than” ni “… dan … kam” (ayirish) bilan adashtirish. “is less than” — taqqoslash, “less than” (is’siz) — ayirish.

b ning ikki baravaridan 4 taga kam

“k less than X” — X dan k ni ayirish. Tartib teskari: avval X, keyin −k.

“The quantity a is 4 less than twice b.”

a = 2b − 4

⚠ a = 4 − 2b deb yozish. Inglizcha so‘z tartibi bo‘yicha emas, ma’no bo‘yicha yozing.

kamida

Qiymat ko‘rsatilgan songa teng yoki undan katta.

“Each class must have at least 12 students.”

x ≥ 12

⚠ x > 12 deb yozish — 12 ning o‘zi ham ruxsat etilgan.

ko‘pi bilan

Qiymat ko‘rsatilgan songa teng yoki undan kichik.

“The number n is at most 20.”

n ≤ 20

⚠ n ≥ 20 deb yozish. “At most” — yuqori chegara.

dan ko‘p emas

“At most” bilan bir xil ma’no: teng yoki kichik.

“A box can hold no more than 50 pounds.”

w ≤ 50

⚠ “no more” ni inkor deb o‘ylab, w > 50 yoki w < 50 yozish.

dan kam emas

“At least” bilan bir xil: teng yoki katta.

“The temperature was no less than 30 degrees.”

T ≥ 30

⚠ T > 30 deb yozish — 30 ham mumkin.

dan kichik yoki teng

Noqat’iy tengsizlik ≤.

“The cost, c, is less than or equal to $80.”

c ≤ 80

⚠ < belgisini ishlatish.

… ning ikki baravari

Bir miqdor boshqasining 2 barobari.

“Maria has twice as much money as Ken.”

M = 2K

⚠ K = 2M deb yozish — kimning puli ko‘pligini tekshiring.

… ning yarmi

Miqdorni 2 ga bo‘lish yoki ½ ga ko‘paytirish.

“Half of the students chose algebra.”

s/2

⚠ “half of x more than y” ni (x+y)/2 deb yozish; to‘g‘risi x/2 + y.

… orasidagi ayirma

Birinchi aytilgan miqdordan ikkinchisini ayirish.

“The difference between x and 5 is 12.”

x − 5 = 12

⚠ 5 − x = 12 deb yozish.

… ning yig‘indisi

Qo‘shish natijasi.

“The sum of two consecutive integers is 41.”

n + (n + 1) = 41

⚠ “the sum of twice x and 3” ni 2(x + 3) deb yozish; to‘g‘risi 2x + 3.

… ning ko‘paytmasi

Ko‘paytirish natijasi.

“The product of x and 4 is 28.”

4x = 28

⚠ “product” ni “mahsulot” deb tarjima qilish — matematikada bu ko‘paytma.

… ning bo‘linmasi

Bo‘lish natijasi; birinchi son bo‘linuvchi.

“The quotient of y and 3 is 7.”

y/3 = 7

⚠ 3/y = 7 deb yozish.

… ga oshirildi

Qo‘shish. Foiz bilan kelsa — ko‘paytuvchi (1 + p/100).

“A number increased by 6 is 20.”

x + 6 = 20

⚠ “increased by 20%” ni x + 20 deb yozish; to‘g‘risi 1.2x.

… ga kamaytirildi

Ayirish.

“A number is decreased by 8 and the result is 15.”

x − 8 = 15

⚠ 8 − x = 15 deb yozish.

… ning nisbati

Birinchi aytilgan miqdor surat (yuqori)da.

“The ratio of boys to girls is 3 to 5.”

b/g = 3/5

⚠ Tartibni almashtirib g/b = 3/5 yozish.

… ga proporsional

To‘g‘ri proporsionallik: y = kx, k — o‘zgarmas.

“The cost is proportional to the number of pounds purchased.”

C = kp

⚠ y = x + k deb yozish — bu proporsionallik emas.

… bilan to‘g‘ri proporsional o‘zgaradi

“Proportional to” ning sinonimi.

“Distance varies directly with time at a constant speed.”

d = kt

⚠ Teskari proporsionallik (y = k/x) bilan adashtirish.

ketma-ket butun sonlar

Bir-biridan 1 ga farq qiladigan butun sonlar. Ketma-ket juft/toq sonlar 2 ga farq qiladi.

“The sum of three consecutive integers is 72.”

n + (n+1) + (n+2) = 72

⚠ Ketma-ket toq sonlarni n, n+1, n+2 deb olish; to‘g‘risi n, n+2, n+4.

butun yechimlar

Tenglama yoki tengsizlikni qanoatlantiradigan faqat butun qiymatlar.

“How many integer solutions does 2 < x ≤ 6 have?”

x ∈ {3, 4, 5, 6} ⇒ 4

⚠ Chegaraviy qiymatni noto‘g‘ri sanash (2 ni qo‘shib yuborish).

x orqali (ifodalang)

Javobni x qatnashgan ifoda ko‘rinishida yozish; boshqa o‘zgaruvchini yolg‘iz qoldirish.

“If 3x + y = 10, what is y in terms of x?”

y = 10 − 3x

⚠ x ni topishga urinish. Savol x ni emas, y ni x orqali so‘rayapti.

mos ravishda

Ro‘yxatlardagi elementlar tartib bo‘yicha bir-biriga mos keladi.

“Pens and notebooks cost $2 and $4, respectively.”

p = 2, n = 4

⚠ Har ikkisi ham 2 yoki 4 dollar deb o‘ylash yoki tartibni almashtirish.

qolgan miqdor

Umumiy miqdordan ishlatilgan qismni ayirish natijasi.

“After spending $x of $60, the remaining amount is $18.”

60 − x = 18

⚠ Qolgan miqdorni ishlatilgan miqdor bilan almashtirish.

x ning qiymati

x o‘rniga qo‘yiladigan aniq son.

“If 2x − 3 = 11, what is the value of x?”

x = 7

⚠ “the value of 2x” so‘ralganda x ni javob qilib berish.

… ifoda (qavs ichida)

Ko‘pincha qavsni bildiradi: “the quantity x + 2” — butun ifoda bitta birlik.

“Five times the quantity x minus 1 equals 20.”

5(x − 1) = 20

⚠ Qavssiz 5x − 1 deb yozish.

eng katta mumkin bo‘lgan qiymat

Shartlarni buzmaydigan eng katta qiymat.

“If 4x + 3 ≤ 27, what is the greatest possible value of x?”

x ≤ 6 ⇒ 6

⚠ Qat’iy tengsizlikda chegaraning o‘zini javob qilib berish.

eng kichik qiymat

Funksiya yoki ifoda erishadigan eng kichik qiymat. Parabola uchun — uchining y koordinatasi (a > 0).

“What is the minimum value of f(x) = (x − 3)² + 5?”

5

⚠ Eng kichik qiymat o‘rniga uning x koordinatasini (3) javob qilish.

eng katta qiymat

Ifoda erishadigan eng katta qiymat. a < 0 parabolada — uchning y koordinatasi.

“The function h(t) = −16t² + 64t has a maximum value at t = 2.”

h(2) = 64

⚠ Qaysi o‘lchov so‘ralganini (t yoki h) aralashtirish.

tenglamani qanoatlantiradi

Qiymat qo‘yilganda tenglik to‘g‘ri bo‘ladi.

“Which ordered pair (x, y) satisfies the equation y = 2x + 1?”

(2, 5)

⚠ Faqat x ni tekshirib, y ni tekshirmaslik.

haqiqiy yechimga ega emas

Kvadrat tenglamada diskriminant manfiy: b² − 4ac < 0.

“For what value of c does x² + 4x + c = 0 have no real solutions?”

16 − 4c < 0 ⇒ c > 4

⚠ “No solution” ni “yechim 0” deb tushunish.

ekvivalent (teng kuchli) ifoda

Barcha x qiymatlarida bir xil natija beradigan ifoda.

“Which expression is equivalent to 3(x + 4) − 2x?”

x + 12

⚠ Faqat bitta x qiymatida tekshirib, ekvivalent deb qaror qilish.