2. Operations (N)
  3. Lesson 6: Subtracting Integers

Answers and Solutions


    1. \(5-2=3\)
      I have \($5\) and I spend \($2\), meaning that I still have \($3\).
    2. \(9-12=(-3)\)
      I have \($9\) and I spend \($12\), creating \($3\) worth of debt.
    3. \((-3)-5=(-8)\)
      I have \($3\) worth of debt and I spend another \($5\), meaning that my total debt is now at \($8\).
    4. \((-7) - (-3)=(-4)\)
      I have \($7\) worth of debt and I remove \($3\) worth of debt.  The result is that I now have \($4\) worth of debt.
    1. negative, \(3\)
    2. negative, \(8\)
    3. negative, \(75\)
    4. negative, \(3\)
    5. positive, \(3\)
  3. These two integers are equal.
  4. We know the answer is \((-6)\), and so we are trying to solve a problem of the form\[\boxed{\phantom\square} - \boxed{\phantom\square} = (-6)\] Since subtraction is not commutative, we have \(2\) separate scenarios to consider.
    • If \((-9)\) is the first integer, we have\[(-9) - \boxed{\phantom\square} = (-6)\] and working backwards, we find the missing integer is \((-3)\).
    • If \((-9)\) is the second integer, we have\[\boxed{\phantom\square} - (-9) = (-6)\] and working backwards, we find the missing integer is \((-15)\).
  5. We need to find the difference between the highest point and the lowest point.
    \(\begin{align*} \text{Height of Everest } - \text{ Depth of Trench} &= 8848 - (-10~994) \\ &= 8848 + 10~994 \\ &= 19~842 \end{align*}\)
    Therefore, the elevation change is \(19~842\) metres.
    1. Show the range of numbers on the number line.
      The greatest difference between two numbers is the same as finding the longest line that you can draw between two numbers, one from each set on the number line.  This occurs when we draw a line from the largest possible integer, \((-64)\), to the smallest possible integer, \((-356)\).  The difference is\[\begin{align*} (-64) - (-356) &= (-64) + 356 \\ &= 292 \end{align*}\] Note that we subtract the smaller number from the larger number to get the greatest difference.
    2. Show the range of numbers on the number line.
      The least difference between two numbers is the same as finding the shortest line that you can draw between two numbers, one from each set on the number line.  This occurs when we draw a line from the two numbers that are closest to one another.  The difference is\[\begin{align*} (-234) - (-120) &= (-234) + 120 \\ &= (-114) \end{align*}\] Note that we subtract the larger number from the smaller number to create the least difference.
  7. Sign of First Integer \(-\) Sign of Second Integer Value Compared to First Integer
    positive \(-\) positive less than integer 1
    positive \(-\) negative greater than integer 1
    negative \(-\) positive less than integer 1
    negative \(-\) negative greater than integer 1
  8. When subtraction is replaced by addition of the opposite integer, the opposite of the second term becomes the first term.  Commuting subtraction does not produce the same result.  Consider the following two examples.
    Example Commuting Subtraction Adding the Opposite
    \(9-14\) \(14-9 = 5\) \((-14)+9 = (-5)\)
    \(12-5\) \(5-12 = (-7)\) \((-5)+12=7\)

    Notice the quantities in the second and third columns are not the same.  This supports the statement that commuting subtraction is not the same as adding the opposite.  Since we know the entries in the third column are correct, it must be the case that subtraction is not commutative.
    Next, we note that rewriting \(9-14\) as \((-14)+9\) does not contradict the statement that "subtraction is not commutative" because to do this, we are using the fact that addition is commutative.  Below we show the intermediate step:

    \[\begin{align*} 9 - 14 &= 9 + (-14) \\ &= (-14) + 9 \end{align*}\]

    Here, we use the property of addition being commutative, not subtraction, and therefore no contradiction has been made.