Algebra Know-It-ALL

Question 4-3

Suppose we have two integers c and d. How can we define c−d on a vertical number line
where values increase as we move upward?

Answer 4-3

We start by finding the point on the line that corresponds to c. Then we move downward
along the line for a distance of d units. We end up at the point for c−d.

Question 4-4

How do signs work when adding and subtracting positive and negative integers?

Answer 4-4

When we add a positive or subtract a negative, the result grows larger. When we subtract a
positive or add a negative, the result grows smaller. For any two integers p and q,

p+ (−q)=p−q

and

p− (−q)=p+q

Question 4-5

Suppose that we start with −6, add −8 to it, then subtract 12 from that, then subtract −5 from
that, then add −2 to that, and finally subtract −23 from that. What’s the result?

Answer 4-5

Let’s work this through in steps, paying careful attention to signs and using parentheses when
we need them:

− 6 + (−8)=− 6 − 8 =− 14

− 14 − 12 =− 26

− 26 − (−5)=− 26 + 5 =− 21

− 21 + (−2)=− 21 − 2 =− 23

− 23 − (−23)=− 23 + 23 = 0

Question 4-6

What does the commutative law tell us about the sum of two integers? What does the associa-
tive law tell us about the sum of three integers? What do these two laws, taken together, allow
us to do?

Answer 4-6

The commutative law tells us that when we add two integers, we can do it in either order and
the sum will be the same. If a and b are integers, then

a+b=b+a

Part One 151