Symbolic Operators

Since \(-20 \blacktriangle 9=-20\), the equation in the problem can be rewritten as \[\begin{align} (-20 \blacktriangle 9) \blacktriangledown (k \blacktriangle 5) &=5\\ -20 \blacktriangledown (k \blacktriangle 5) &=5. \end{align}\] Now, the value of the left-hand side must be either \(-20\) or \((k \blacktriangle 5)\) by the definition of \(\blacktriangledown\). Since the value on the right-hand side is \(5\), which is not \(-20\), it must be the case that \((k \blacktriangle 5) =5\). Also, note that \( (-20) \blacktriangledown 5 = 5\) since \(\lvert - 20 \rvert > 5\). Therefore, our goal is to find the number of integers \(k\) that satisfy \[k \blacktriangle 5 =5. \qquad (1) \]

If \( \vert k \rvert>5\), then \(k \blacktriangle 5 =k \), which does not satisfy \((1)\).

If \(k=-5\), then \(k \blacktriangle 5= -5 \blacktriangle 5 =-5\), which does not satisfy \((1)\).

If \(k=5\), then \(k \blacktriangle 5= 5 \blacktriangle 5 =5\), which satisfies \((1)\).

If \( \lvert k \rvert<5\), then \(k \blacktriangle 5 =5\), which satisfies \((1)\).

It follows that the integers \(k\) that satisfy \((1)\) lie in the interval \([-4, 5]\). Therefore, the number of integers \(k\) satisfying the equation is \[5-(-4)+1=10.\]