Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
5
Expressions Containing Signed Numbers
Add, subtract, multiply, and divide positive and negative numbers
1.11 Simplify the expression: 16 + (–9).
This expression contains adjacent or “double” signs, two signs next to one
another. To simplify this expression, you must convert the double sign into a
single sign. The method is simple: If the two signs in question are different,
replace them with a single negative sign; if the signs are the same (whether both
positive or both negative), replace them with a single positive sign.
In this problem, the adjacent signs are different, “+ –,” so you must replace them
with a single negative sign: –.
1.12 Simplify the expression: –5 – (+6).
This expression contains the adjacent signs “– +.” As explained in Problem 1.11,
the double sign must be rewritten as a single sign. Because the adjacent signs
are different, they must be replaced with a single negative sign.
–5 – (+6) = –5 – 6
To simplify the expression –5 – 6, or in fact any expression that contains
signed numbers, think in terms of payments and debts. Every negative number
represents money you owe, and every positive number represents money you’ve
earned. In this analogy, –5 – 6 would be interpreted as a debt of $5 followed by
a debt of $6, as both numbers are negative. Therefore, –5 – 6 = –11, a total debt
of $11.
1.13 Simplify the expression: 4 – (–5) – (+10).
This expression contains two sets of adjacent or “double” signs: “– –” between
the numbers 4 and 5 and “– +” between the numbers 5 and 10. Replace like
signs with a single + and unlike signs with a single –.
4 – (–5) – (+10) = 4 + 5 – 10
Simplify the expression from left to right, beginning with 4 + 5 = 9.
4 + 5 – 10 = 9 – 10
Some algebra
books write positive
and negative signs
higher and smaller, like
this: 16 +
–
9. I’m sorry, but
that’s just weird. It’s
perfectly ne to turn
that teeny oating
sign into a regular
sign: 16 + –9.
Think of it
this way. If the
two signs agree with
each other (if they’re
both positive or both
negative), then that’s a
good thing, a POSITIVE
thing. On the other hand,
when two signs can’t
agree with each other
(one’s positive and one’s
negative), then that’s
no good. That’s
NEGATIVE.
There’s one other technique you can use to add and subtract
signed numbers. If two numbers have different signs (like 9 and –10), then subtract
them (10 – 9 = 1) and use the sign from the bigger number (10 > 9, so use the negative sign
attached to the 10 to get –1 instead of 1). If the signs on the numbers are the same, then
add the numbers together and use the shared sign. In other words, to simplify –12
– 4, add 12 and 4 to get 16 and then stick the shared negative sign
out front: –16.
Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
6
To simplify 9 – 10 using the payments and debts analogy from Problem 1.12, 9
represents $9 in cash and –10 represents $10 in debts. The net result would be a
debt of $1, so 9 – 10 = –1.
1.14 Simplify the expression: .
Choosing the sign to use when you multiply and divide numbers works very
similarly to the method described in Problem 1.11 to eliminate double signs.
When two numbers of the same sign are multiplied, the result is always positive.
If, however, you multiply two numbers with different signs, the result is always
negative.
In this case, you are asked to multiply the numbers 6 and –3. Because one is
positive and one is negative (that is, their signs are different), the result must be
negative.
1.15 Simplify the expression: .
When signed numbers are divided, the sign of the result once again depends
upon the signs of the numbers involved. If the numbers have the same sign, the
result will be positive, and if the numbers have different signs, the result will be
negative. In this case, both of the numbers in the expression, –16 and –2, have
the same sign, so the result is positive: .
1.16 Simplify the expression: (3)( –3)(4)(–4).
Multiply the signed numbers in this expression together working from left to
right. In this way, because you are multiplying only two numbers at a time, you
can apply the technique described in Problem 1.14 to determine the sign of
each result. The leftmost two numbers are 3 and –3; they have different signs, so
multiplying them together results in a negative number: (3)( –3) = –9.
(3)( –3)(4)( –4) = (–9)(4)( –4)
Again multiply the two leftmost numbers. The signs of –9 and 4 are different, so
the result is negative: (–9)(4) = –36.
(–9)(4)( –4) = (–36)( –4)
The remaining signed numbers are both negative; because the signs match,
multiplying them together results in a positive number.
(–36)( –4) = 144
You could
also write
,
but you don’t HAVE
to write a + sign in
front of a positive
number. If a number
has no sign in
front of it, that
means it’s
positive.
There’s no
multiplication
sign written
between (3) and
(–3), so how did
you know to multiply
them together? It’s
an “unwritten rule”
of algebra. When two
quantities are written
next to one another
and no sign separates
them, multiplication is
implied. That means
things like 4(9), 10y,
and xy are all
multiplication
problems.
Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
7
1.17 Simplify the expression: .
The straight lines surrounding 9 in this expression represent an absolute value.
Evaluating the absolute value of a signed number is a trivial matter—simply
make the signed number within the absolute value bars positive and then
remove the bars from the expression. In this case, the number within the
absolute value notation is already positive, so it remains unchanged.
You are left with two signed numbers to combine: +4 and –9. According to the
technique described in Problem 1.11, combining $4 in assets with $9 in debt has
a net result of $5 in debt: 4 – 9 = –5.
1.18 Simplify the expression: .
The absolute value of a negative number, in this case –10, is the opposite of the
negative number: .
1.19 Simplify the expression: .
If this problem had no absolute value bars and used parentheses instead, your
approach would be entirely different. The expression –(5) – (–5) has the
double sign “– –,” which should be eliminated using the technique described in
Problems 1.11–1.13. However, absolute value bars are treated differently than
parentheses, so this expression technically does not contain double signs. Begin
by evaluating the absolute values: and .
Absolute
value bars
are the anti-
depressants of the
mathematical world.
They make everything
inside positive. To say
that more precisely, they
take away the negative
of the number inside.
That means
.
However, the mood-
altering lines have no
effect on positive
numbers:
.
Absolute
values are
simple when
there’s only one
number inside. If
the number inside
is negative, make it
positive and drop the
absolute value bars.
If the number’s
already positive,
leave it alone
and just drop
the bars.
Well, it
doesn’t contain
double signs YET.
It will in just a
moment.
See? There’s
the double sign.
When
turned into
(+5), the negative sign
in front of the absolute
values didn’t go away.
In the next step, you
eliminate the double
sign “– +” to get
–5 – 5.
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