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Difficulty: Very hardOrder of Operations and Number Properties

For all non-zero real numbers xx and yy, let the operation ⋆\star be defined by x⋆y=xy−yxx \star y = \frac{x}{y} - \frac{y}{x}. Which of the following equations must be true for all non-zero real numbers aa, bb, and cc?

  1. A
    a⋆b=b⋆aa \star b = b \star a
  2. B
    a⋆(b⋆c)=(a⋆b)⋆ca \star (b \star c) = (a \star b) \star c
  3. C
    a⋆(b+c)=(a⋆b)+(a⋆c)a \star (b + c) = (a \star b) + (a \star c)
  4. a⋆b=−(b⋆a)a \star b = -(b \star a)Answer
  5. E
    (a+b)⋆c=(a⋆c)+(b⋆c)(a + b) \star c = (a \star c) + (b \star c)

Answer

The equation stating that the operation on two numbers is equal to the negative of the operation on those numbers in reverse order, which is a⋆b=−(b⋆a)a \star b = -(b \star a)
Applying the definition of the custom operation, the expression for b⋆ab \star a is ba−ab\frac{b}{a} - \frac{a}{b}. Factoring out −1-1 from this expression yields −(ab−ba)-\left(\frac{a}{b} - \frac{b}{a}\right), which is equivalent to −(a⋆b)-(a \star b). Multiplying both sides by −1-1 results in the equation a⋆b=−(b⋆a)a \star b = -(b \star a), which must be true for all non-zero real numbers aa and bb.

Step-by-Step Solution

1
Evaluate the expression for b⋆ab \star a using the custom operation definition.
b⋆a=ba−abb \star a = \frac{b}{a} - \frac{a}{b}
To analyze the result of reversing the order of operands.
2
Factor out a negative sign from the right-hand side of the evaluated expression.
b⋆a=−(−ba+ab)=−(ab−ba)b \star a = -\left(-\frac{b}{a} + \frac{a}{b}\right) = -\left(\frac{a}{b} - \frac{b}{a}\right)
To align the expression's terms with the definition of a⋆ba \star b.
3
Substitute the definition of a⋆ba \star b back into the expression.
b⋆a=−(a⋆b)b \star a = -(a \star b)
To establish the relationship between the two permutations of the operation.
4
Negate both sides of the equation to find the equivalent representation.
a⋆b=−(b⋆a)a \star b = -(b \star a)
To match the final equation with the options.

Key Concept

Algebraic properties of custom operations and order of operations

Alternative Method

Instead of algebraic proof, substitute small, distinct non-zero integers (e.g., a=2a = 2, b=1b = 1, c=3c = 3) to evaluate the expressions in each option, enabling the elimination of equations that do not hold.
Estimated Time:1m 30s
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Topics

Order of Operations and Number Properties

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