Consider the arithmetic operations ▲ and ▼ defined by
a ▲ b a ▼ b = ⎩ ⎨ ⎧ a b if ∣ a ∣ ≥ ∣ b ∣ if ∣ a ∣ < ∣ b ∣ = ⎩ ⎨ ⎧ a b if ∣ a ∣ ≤ ∣ b ∣ if ∣ a ∣ > ∣ b ∣
How many integers k are there such that ( − 2 0 ▲ 9 ) ▼ ( k ▲ 5 ) = 5 ?
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Since − 2 0 ▲ 9 = − 2 0 , thus the equation in the problem can be rewritten as ( − 2 0 ▲ 9 ) ▼ ( k ▲ 5 ) = − 2 0 ▼ ( k ▲ 5 ) = 5 . Considering the final equality − 2 0 ▼ ( k ▲ 5 ) = 5 , the value of the left-hand side must be either − 2 0 or ( k ▲ 5 ) by the definition of ▼ . Since the value on the right-hand side is 5 , which is not − 2 0 , it must be the case that ( k ▲ 5 ) = 5 . So, our goal now becomes finding the number of integers k that satisfy k ▲ 5 = 5 . ( 1 )
If ∣ k ∣ > 5 , then k ▲ 5 = k , which contradicts ( 1 ) .
If k = − 5 , then k ▲ 5 = − 5 ▲ 5 = − 5 , which contradicts ( 1 ) .
If k = 5 , then k ▲ 5 = 5 ▲ 5 = 5 , which satisfies ( 1 ) .
If ∣ k ∣ < 5 , then k ▲ 5 = 5 , which satisfies ( 1 ) .
Thus, the integers k that satisfy ( 1 ) lie in the interval [ − 4 , 5 ] . Therefore, the number of integers in this interval is 5 − ( − 4 ) + 1 = 1 0 .