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ACT Exam  >  ACT Questions  >  For all numbers x and y, let the operation n ... Start Learning for Free
For all numbers x and y, let the operation n be defined as x € y = 2xy - 4x. If a and b are positive integers, which of the following can be equal to zero?
I. a € b
II. (a - b) € b
III. b € (a - b)
  • a)
    I only
  • b)
    II only
  • c)
    III only
  • d)
    I and II only
  • e)
    I, II, and III
Correct answer is option 'E'. Can you explain this answer?
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For all numbers x and y, let the operation n be defined as x € y ...
The expression "x n y" is not clearly defined in the prompt. Please provide more information or clarification about the operation "n" in order for me to continue.
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For all numbers x and y, let the operation n be defined as x € y ...
To solve this problem, substitute a for x and b for y and set it equal to zero: 2ab - 4a = 0. Factor out 2a: 2a(b - 2) = 0. Since a cannot equal zero (it must be positive), then (b - 2) = 0. Solving for b gives you b = 2. Since b is an integer, Roman Numeral I can equal 0. Eliminate answer choices B and C. Substitute (a - b) for x and b for y and set it equal to zero: 2(a - b)b - 4(a - b) = 0. Factor out 2(a - b): 2(a - b)(b - 2) = 0. This is true if a = b or if b = 2. Since a or b can be an integer, Roman Numeral II can equal 0. Eliminate answer choice A. Substitute b for x and (a - b) for y and set it equal to zero: 2b(a - b) - 4b = 0. Factor out 2b: 2b[(a - b) - 2] = 0. Since b cannot equal zero (it must be positive), then (a - b - 2) = 0. Solving for a gives you a = b + 2, which can be an integer. Solving for b gives you b = 2 - a, which can be an integer. So, Roman Numeral III can equal 0. Eliminate answer choice D. Each of the Roman Numerals can be equal to zero.
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Both gases and liquids are considered to be fluids that have individual molecules that move around with kinetic and potential energy. Kinetic energy, defined as the energy related to motion, takes three forms: translational energy that occurs as a molecule moves from position A to position B, rotational energy that occurs as a molecule spins around an imaginary axis at its center of mass, and vibrational energy that occurs as individual atoms in a molecular bond move towards and away from each other. Usually, molecules possess varying combinations of kinetic energy forms. In contrast, potential energy is defined as stored energy that could be released to become kinetic energy. The total energy of a molecule is fixed, meaning that a molecule has some combination of kinetic and potential energies.Varying amount of kinetic and potential energies define how molecules in a fluid interact with each other. For example, when the kinetic energy of a molecule is high (greater than 1000J), it can no longer interact with neighboring molecules strongly enough to remain a liquid. However, if the potential energies are too high (greater than 1000 J), molecules cannot escape a liquid to become a gas. If the kinetic energy is high and the potential energy is low, molecules tend to become a gas and can be modeled by an equation known as the Ideal Gas Law:PV=nRT Where P is the pressure of a gas, V is the volume, n is the number of moles of a gas, R is a constant, and T is temperature in degrees Kelvin.The Ideal Gas Law perfectly applies to particles with no mass, no intermolecular interactions, and no true volume. However, real molecules do not adhere perfectly to the Ideal Gas Law.Q.An oxygen molecule moving from the left side of a chamber to the right would display what type(s) of kinetic energy?

Both gases and liquids are considered to be fluids that have individual molecules that move around with kinetic and potential energy. Kinetic energy, defined as the energy related to motion, takes three forms: translational energy that occurs as a molecule moves from position A to position B, rotational energy that occurs as a molecule spins around an imaginary axis at its center of mass, and vibrational energy that occurs as individual atoms in a molecular bond move towards and away from each other. Usually, molecules possess varying combinations of kinetic energy forms. In contrast, potential energy is defined as stored energy that could be released to become kinetic energy. The total energy of a molecule is fixed, meaning that a molecule has some combination of kinetic and potential energies.Varying amount of kinetic and potential energies define how molecules in a fluid interact with each other. For example, when the kinetic energy of a molecule is high (greater than 1000J), it can no longer interact with neighboring molecules strongly enough to remain a liquid. However, if the potential energies are too high (greater than 1000 J), molecules cannot escape a liquid to become a gas. If the kinetic energy is high and the potential energy is low, molecules tend to become a gas and can be modeled by an equation known as the Ideal Gas Law:PV=nRT Where P is the pressure of a gas, V is the volume, n is the number of moles of a gas, R is a constant, and T is temperature in degrees Kelvin.The Ideal Gas Law perfectly applies to particles with no mass, no intermolecular interactions, and no true volume. However, real molecules do not adhere perfectly to the Ideal Gas Law.Q. As the kinetic energy of a molecule increases, one would expect the potential energy to

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For all numbers x and y, let the operation n be defined as x € y = 2xy - 4x. If a and b are positive integers, which of the following can be equal to zero?I. a € bII. (a - b) € bIII. b € (a - b)a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II, and IIICorrect answer is option 'E'. Can you explain this answer?
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For all numbers x and y, let the operation n be defined as x € y = 2xy - 4x. If a and b are positive integers, which of the following can be equal to zero?I. a € bII. (a - b) € bIII. b € (a - b)a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II, and IIICorrect answer is option 'E'. Can you explain this answer? for ACT 2025 is part of ACT preparation. The Question and answers have been prepared according to the ACT exam syllabus. Information about For all numbers x and y, let the operation n be defined as x € y = 2xy - 4x. If a and b are positive integers, which of the following can be equal to zero?I. a € bII. (a - b) € bIII. b € (a - b)a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II, and IIICorrect answer is option 'E'. Can you explain this answer? covers all topics & solutions for ACT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For all numbers x and y, let the operation n be defined as x € y = 2xy - 4x. If a and b are positive integers, which of the following can be equal to zero?I. a € bII. (a - b) € bIII. b € (a - b)a)I onlyb)II onlyc)III onlyd)I and II onlye)I, II, and IIICorrect answer is option 'E'. Can you explain this answer?.
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