Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
8
1.20 Simplify the expression: .
Do not eliminate double signs in this expression until you have first addressed
the absolute values.
Combine the signed numbers two at a time, working from left to right. Begin
with 2 – 7 = –5.
1.21 Simplify the expression: .
This problem contains the absolute value of an entire expression, not just a single
number. In these cases, you cannot simply remove the negative signs from each
term of the expression, but rather simplify the expression first and then take the
absolute value of the result.
To simplify the expression 3 + (–16) – (–9), you must eliminate the double signs
and them combine the numbers one at a time, from left to right.
Grouping Symbols
When numbers band together, deal with them rst
1.22 Simplify the expression: .
When portions of an expression are contained within grouping symbols—like
parentheses (), brackets [], and braces {}—simplify those portions of the
expression first, no matter where in the expression it occurs. In this expression,
is contained within parentheses, so multiply those numbers: .
1.23 Simplify the expression: .
The only difference between this expression and Problem 1.22 is the placement
of the parentheses. This time, the expression 7 + 10 is surrounded by grouping
symbols and must be simplified first.
–5 + 5 = 0
For now, the
parentheses and
other grouping
symbols will tell
you what pieces of
a problem to simplify
rst. When parentheses
aren’t there to help, you
have to apply something
called the “order of
operations,” which
is covered in
Problems 3.30–
3.39.
Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
9
By comparing this solution to the solution for Problem 1.22, it is clear that the
placement of the parentheses in the expression had a significant impact on the
solution.
1.24 Simplify the expression: .
Although this expression contains parentheses and brackets, the brackets are
technically the only grouping symbols present; the parentheses surrounding –11
are there for notation purposes only. Simplify the expression inside the brackets
first.
1.25 Simplify the expression: .
This expression contains two sets of nested grouping symbols, brackets and
parentheses. When one grouped expression is contained inside another,
always simplify the innermost expression first and work outward from there.
In this case, the parenthetical expression should be simplified first.
A grouped expression still remains in the expression, so it must be simplified
next.
1.26 Simplify the expression: .
Grouping symbols are not limited to parentheses, brackets, and braces. Though
it contains none of the aforementioned elements, this fraction consists of two
grouped expressions. Treat the numerator (6 + 10) and the denominator (14
– 8) as individual expressions and simplify them separately.
Double signs, like
in the expression
19 + (–11), are ugly
enough, but it’s just
too ugly to write the
signs right next to each
other like this:
19 + – 11. If you look
back at Problems
1.11–1.13, you’ll notice
that the second
signed number is
always encased
in parentheses if
leaving them out
would mean two
signs are
touching.
“Nested”
means that
one expression
is inside the
other one. In this
case,
is
nested inside the
bracketed expression
because
the expression inside
parentheses is also
inside the brackets.
Nested expressions are
like those egg-shaped
Russian nesting dolls.
You know the ones?
When you open
one of the dolls,
there’s another,
smaller one
inside?
If you’re not sure how
turned
into
, you divide the numbers in the top and bottom
of the fraction by 2:
and . That process
is called “simplifying” or “reducing” the fraction and is
explained in Problems 2.11–2.17.
Chapter One — Algebraic Fundamentals
The Humongous Book of Algebra Problems
10
1.27 Simplify the expression: .
Like Problem 1.26, this fractional expression has, by definition, two implicit
groups, the numerator and the denominator. However, it contains a second
grouping symbol as well, absolute value bars. The absolute value expression is
nested within the denominator, so simplify the innermost expression, first.
Now simplify the numerator and denominator separately.
Any number divided by itself equals 1, so = 1, but note that the numerator
is negative. According to Problem 1.15, when numbers with different signs are
divided, the result is negative.
1.28 Simplify the expression: .
This expression consists of two separate absolute value expressions that are
subtracted. The left fractional expression requires the most attention, so begin
by simplifying it.
Now that the fraction is in a more manageable form, determine both of the
absolute values in the expression.
1.29 Simplify the expression: .
This problem contains numerous nested expressions—braces that contain
brackets that, in turn, contain parentheses that include an absolute value.
Begin with the innermost of these, the absolute value expression.
The innermost expression surrounded by grouping symbols is now (3 + 1), so
simplify it next.
The “numerator”
is the top part of the
fraction and the
“denominator” is
the bottom part.
According to
the end of Problem
1.27, when you divide
a number and its
opposite (like 7 and
–7), you get –1.
Three if you
don’t count
as a group
(because it has
only one number
inside). Four if
you do count
it.
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