We're asked to find the difference. We have eighty-two minus ninety-three. Because ninety-three is greater than eighty-two, let's write this
difference as a sum. We can do this
because subtracting the integer is the same
as adding the opposite. So because subtracting b is the same as adding negative
b, subtracting ninety-three is the same as adding negative ninety-three. So we can write eighty-two minus ninety-three as eighty-two
plus negative ninety-three. Now looking at this sum, we should be able to recognize that the
negative ninety-three is going to outweigh the positive eighty-two, and therefore, this sum is going
to be negative. Now let's find this sum,  which equals this difference using the formal
rules for adding integers. And then we'll also show this sum on the number line. So
because we have a positive plus a
negative, we'll be using the rules here
for adding two integers with different signs. So step one, we find the
absolute value of both integers, so we need to find the absolute value of positive eighty-two, and the
absolute value of negative ninety-three. The absolute number of a numbers is the numbers distance
from zero, and distance is always positive. And therefore, the absolute value
of positive eighty-two is positive eighty-two. The absolute value of negative ninety-three is positive ninety-three. Step two, we
subtract the smaller value from the
larger value, so we have ninety-three minus eighty-two, which
equals eleven. And then step three, we need to determine the sign of the sum. The sign of the sum
is the original sign of the number with the larger absolute value. So because
negative ninety-three had the larger absolute value,  this sum is not eleven, but its negative eleven.
Again, it's negative eleven because negative ninety-three had the larger absolute value. So this sum equals
negative eleven, which also equals eighty-two minus ninety-three.
Let's also show this sum on the number line. We'll begin by modeling eighty-two on the
number line, so we start at zero and
move right eighty-two units
to eighty-two.
And now from here, because we're adding negative ninety-three, we need to move left ninety-
three units. So starting here at positive
eighty-two, notice how if we move left eighty-two units,
we'd be back at zero. We want to move ninety-three units to the left,
so we need to go eleven more units to the left to negative eleven.
Eighty-two plus negative ninety-three, which equals eighty-two minus ninety-three equals negative eleven.
I hope you found this helpful.
