SAT Math Multiple Choice Question 643: Answer and Explanation

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Question: 643

13. For all numbers x and y, let z be defined by the equation z = |22 - x2 - y2| + 22. What is the smallest possible value of z?

  • A. 0
  • B. 4
  • C. 8
  • D. 16

Correct Answer: B

Explanation:

B

Algebra (absolute values) MEDIUM-HARD

In order to minimize the value of |22 - x2 - y2| + 22, we must minimize the absolute value. But the least possible value of any absolute value expression is 0, so we must ask: is it possible for the expression inside the absolute value operator to equal 0? A little trial and error should reveal that it can if, for instance, x = 2 and y = 0. Notice that this gives us |2 - 22 - 02| + 2 = |0| + 22 = 4. Since the absolute value cannot be less than 0, this must be the minimum possible value.

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