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The number ww is irrational. Which statement about 6−w6 - w is true?\newlineChoices:\newline(A)6−w6 - w is rational.\newline(B)6−w6 - w is irrational.\newline(C)6−w6 - w can be rational or irrational, depending on the value of ww.

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Full solution

Q. The number ww is irrational. Which statement about 6−w6 - w is true?\newlineChoices:\newline(A)6−w6 - w is rational.\newline(B)6−w6 - w is irrational.\newline(C)6−w6 - w can be rational or irrational, depending on the value of ww.
  1. Identify Number 66: Identify the nature of the number 66. 66 is a rational number because it can be expressed as a fraction of two integers (61\frac{6}{1}).
  2. Consider Number ww: Consider the nature of the number ww.ww is given as an irrational number, which means it cannot be expressed as a fraction of two integers.
  3. Analyze Subtraction: Analyze the subtraction of an irrational number from a rational number. The difference between a rational number and an irrational number is always irrational. This is because if 6−w6 - w were rational, then adding ww to both sides would imply that 66 is equal to a rational plus an irrational number, which would make 66 irrational. This is a contradiction because we know 66 is rational.
  4. Conclude Result: Conclude the nature of 6−w6 - w. Since 66 is rational and ww is irrational, their difference 6−w6 - w must be irrational.

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