Exercises

1. Find the product.
   1. \((-5) \times 4\)
   2. \((-12) \times (-3)\)
   3. \(7 \times (-8)\)
   4. \((-6) \times (+3) \times 2\)
   5. \(8 \times \big((-9) + 3\big)\)
2. Find the missing integer.
   1. \((-4) \times \boxed{\phantom\square} = -24\)
   2. \(\boxed{\phantom\square} \times 7 = -56\)
   3. \(\boxed{\phantom\square} \times (-6) = 54\)
3. The product of two integers is \(-20\). What are the two integers? Is there more than one possible answer?
4. Explain why the product of two negative integers is always greater than the sum of the same two integers.
5. Write a different integer in each of the five squares in the image so that the product of each pair of integers is shown on the line that connects them.
   Try to find a second solution.
    
6. Find a pair of integers whose
   1. Sum is \(-5\) and product is \(-24\).
   2. Sum is \(-7\) and product is \(12\).
   3. Sum is \(4\) and product is \(-5\).
   4. Difference is \(7\) and product is \(-10\).
7. A cozy mountain-side hotel has \(60\) rooms. Each day the hotel makes a profit of \($80.00\) for each room that is occupied and loses \($40.00\) for each room that is vacant.
   Use positive and negative integers to write the expression for each of the following situations. Calculate the hotel's net profit using the following equation.

   Number of Occupied Rooms

   \(\Large\boxed{\phantom\square}\)

   \(\times\)

   Profit per Room

   \(\Large\boxed{\phantom\square}\)

   \(+\)

   Number of Vacant Rooms

   \(\Large\boxed{\phantom\square}\)

   \(\times\)

   Loss per Room

   \(\Large\boxed{\phantom\square}\)

   \(=\)

   Net Profit

   \(\Large\boxed{\phantom\square}\)

   1. \(38\) rooms occupied
   2. \(18\) rooms occupied
   3. \(10\) rooms vacant
   4. Calculate the number of rooms that need to be occupied so that the net profit is \(0\).
8. A math test consisted of  \(5\) true/false questions. The following equation is used to calculate the total score.

   Total
   Score
   \(\Large\boxed{\phantom\square}\)

   \( =\)

   \(3\)

   \(\times\)

   Number Correct

   \(\Large\boxed{\phantom\square}\)

   \(+\)

   \((-2)\)

   \(\times\)

   Number Incorrect

   \(\Large\boxed{\phantom\square}\)

   \(+\)

   \(0\)

   \(\times\)

   Number Unanswered

   \(\Large\boxed{\phantom\square}\)

   1. What is the maximum quiz score?
   2. What is the minimum quiz score?
   3. Describe a situation where a student scores \(3\) on the quiz.
   4. Describe a situation where a student scores \(-3\) on the quiz.
9. Jane says to Reyansh, "I am thinking of a number, call it \(n\), where \((-n)^2 = -n^2\) is true."
   1. What is Jane's number?
   2. Explain why there is only one number that satisfies \((-n)^2 = -n^2\).
10. The absolute value of a number represents the distance between the number and \(0\) on the number line.
    
    The bar symbol "\(\lvert\)" is placed on either side of a number to mean "absolute value." For example, \(\lvert 3 \rvert=3\) because \(3\) is three units away from \(0\) and \(\lvert-3\rvert=3\) because \(-3\) is also three units away from \(0\).
    1. Simplify the following.
       1. \(\lvert 2 \times 3 \rvert\)
       2. \(\lvert (-4) \times 8 \rvert\)
       3. \(\lvert 5 \times (-6) \rvert\)
       4. \(\lvert (-4) \times (-7) \rvert\)
    2. Find all pairs of integers that satisfy \(\left\lvert m \times n \right \rvert =13\).
    3. How many pairs of non-zero integers satisfy \(\left\lvert m \times n \right \rvert \leq 4\).