rationals integers irrationals · pdf file“multiply by the reciprocal ... its...
TRANSCRIPT
1
You may have counted numbers on your fingers – sometimes I do!
If you can imagine starting with 1 and counting “up” forever, that’s the set of counting
numbers, which some refer to as the set of natural numbers.
If you include the number zero, it is called the set of whole numbers.
If you include the opposites of each counting number, it is called the set of integers.
If you include fractions formed with integers, it is called the set of rational numbers.
If you include numbers that exist yet are not rational (irrationals), it is called the set of real
numbers.
The following diagram illustrates the relationships among the sets described above.
These are the numbers you are expected to have “dealt with” before (computationally).
You should know how to perform the basic operations of arithmetic; namely, addition,
subtraction, multiplication, division and exponents (powers).
The prevailing rule in computation is to work in order of decreasing complexity, which leads
to the hopefully familiar phrase “Order of Operations:”
NATURALS
1, 2, 3, …
WHOLES
INTEGERS
… –3, –2, –1 0
RATIONALS
–11.282345
REAL
NUMBERS
IRRATIONALS
2
The expression “2 + 3 × 4” is evaluated to 14, and not 20, because multiplication precedes
addition.
If the intention is to perform the addition first, parentheses must be used: (2 + 3) × 4 = 20
In the US, the acronym PEMDAS (for Parentheses, Exponentiation, Multiplication, Division,
Addition, Subtraction) is memorized, sometimes along with the mnemonic device
“Please Excuse My Dear Aunt Sally”
Other countries approach it slightly differently. For example, in Australia and Canada, the
acronym BEDMAS is often used as a mnemonic for
Brackets, Exponents, Division, Multiplication, Addition, and Subtraction.
In the UK and New Zealand, the acronym BODMAS is often used for
Brackets, Orders, Division, Multiplication, Addition, Subtraction.
Warning: A very popular error is to ignore the “left-to-right within a group” rule, and always
multiply (M) before dividing (D), or add (A) before subtracting (S).
For example, you need to believe that 273545 , not 3, and that 10 – 3 + 2 = 9, not 5.
This means that all of the acronyms above could be misleading!
Exercises: Simplify.
a. 2517532
b. 2525
1379
2/104
c.
3915
4336
d. 7281623/1584
Solutions: a. 50 b. 9 c. 2 d. 35
Grouping Symbols Powers Products/Quotients Sums/Differences
(Left-to-right within each group)
3
A fraction is a way of expressing a quantity based on an amount that is divided into a
number of equal-sized parts.
For example, each part of a cake split into four equal parts is called one fourth (and
represented numerically as ¼); two fourths is half the cake, and eight fourths would make two
cakes.
A fraction is a ratio and quotient of numbers; for example:
three-fourths = 3:4 = 344/34
34
3
In our cake example, where one fourth is represented numerically as ¼, the bottom number,
called the denominator, is the total number of equal parts making up the cake as a whole,
and the top number, called the numerator, is how many of these parts we wish to represent.
The word “numerator” is related to the word enumerate, meaning to “tell how many;” thus the
numerator tells us how many parts we “have” in the indicated fraction.
To denominate means to “give a name” or “tell what kind;” thus the “denominator” tells us
what kind of parts are available (halves, thirds, fourths, etc.).
The numerator and denominator are often called the “terms” of the fraction.
A fraction is in lowest terms when its numerator and denominator share no common factors
but 1.
Note that because it is impossible to divide something into zero equal parts, zero can never
be the denominator of a fraction.
However, zero divided into any number of parts is still 0, so 00
x for all x ≠ 0.
When the numerator exceeds the denominator, the fraction represents more than one whole,
and is called improper.
In this case, another way of writing that fraction is called mixed-number notation, in which
we represent the improper fraction as the sum of a whole and a “remaining,” proper fraction.
One fourth = ¼ Two fourths = = ½ Eight fourths = = 2
4
The word “remaining” was used since we find mixed number notation using division.
Example: 1172
11
292
7
22
291111
29
r
To convert from mixed-number notation to the improper fraction form, reverse the division
by multiplying, then adding back the remainder:
Examples: 15
1248124412048158154
154
10
7377337037107103
103
Here is a quick, formulaic review of fraction arithmetic:
Multiplication: bd
ac
d
c
b
a “Straight across”
Division: bc
ad
c
d
b
a
d
c
b
a “Multiply by the reciprocal”
Exponentiation: N
NN
b
a
b
a
“Spread the power evenly”
Addition: bd
bcad
bd
bc
bd
ad
d
c
b
a Subtraction:
bd
bcad
bd
bc
bd
ad
d
c
b
a
“Common denominator, then add/subtract numerators”
Exercises: Simplify:
a.
2
7
5
21
20
7
31
b.
38
17
13
93
3
28
Solutions: a. 314 b. 17
216,1
The whole quotient is followed by the remainder as
numerator with the original denominator.
5
Some facts/properties/laws of arithmetic which give instruction and technique…
…and more:
■ The Commutative Property: Order is irrelevant for addition and multiplication.
Symbolically: a + b = b + a and ab = ba
That is, we should agree that: 3 + 7 = 7 + 3 = 10 and 455995
■ The Associative Property: Grouping is irrelevant for addition and multiplication.
Symbolically: cbacba and cabbca
That is, we should agree that:
51172113172113172113 and 168674674674
This could also be stated as, “For addition or multiplication, parentheses are irrelevant”
■ The Distributive Law:
Multiplication distributes equally to all addends of a sum or difference.
Symbolically: acabcba or acabacb or adacabdcba
6
“Algebra: a branch of mathematics in which arithmetic relations are generalized and
explored by using letter symbols to represent numbers, variable quantities or other
mathematical entities, the letter symbols being combined…in accordance with assigned rules.”
-Webster’s Third New International Dictionary, 1961
In this course, we will follow the definition above. It may sound kind of “far-out,” but it is very
real. For example, say I can fill a gallon water jug for 35 cents. To fill J jugs of water would
then cost me JJ 3535 cents. Does that sound silly? Not when I’m finding change to fill
those jugs!
A variable is a symbol that represents a number that may change depending on the
circumstance. Its mathematical usage is similar to that of a pronoun in grammar.
A constant is a symbol that represents a specific, fixed number.
Examples: Many use the variable t to represent time, h to represent height, w for width,
etc. We often use the letter x for the variable when the variable stands for a general number
without context.
A couple of important constants:
Pi = = 3.14159265358… (constant), an important number in many real applications
Euler’s number = e = 2.71828182846… (constant), also an important number in many real
applications
An algebraic expression is a combination of constants and/or variables and/or symbols of
operators.
Examples:
–77 2r ie 123 2 xx 12
12
xx
yy
22 r
r
2222 xb
b
xa
a
An expression must be well-formed; that is, it must “make sense.”
The expression 2 + 3 is well-formed; the expression 9/3212 is not.
Yes, a number is an algebraic expression!
7
Notice that, technically speaking, neither is the expression 104
If the 10 is meant to be negative, parentheses are often used to indicate it: 104
For a given combination of values for the variables, an expression may be evaluated by
substituting the values for the variables and simplifying completely.
For some combinations of values of variables, an expression may be undefined.
Example: y
x , evaluated for x = 10, y = 5, yields 25
10 , but is undefined for y = 0 (right?)
Exercises: Evaluate these expressions for 4 and 5 yx .
a. 3223 324 yyxx b. yxy
yxxy 2512
2
Solutions: a. –108 b. 16
Two expressions are said to be equivalent if, for every combination of values for the
variables, they evaluate equally. What, then, might make one expression “superior” to an
equivalent one? It actually depends on the situation, but in general, SIMPLICITY is a desirable
state.
Therefore the next thing we’ll “do” with expressions is start learning how to simplify them.
What does it mean to simplify?
It means to perform as many operations as possible, in the correct Order.
A correctly simplified expression will be equivalent to the original and any intermediates.
Learning what is “possible” and what is not will take some time, sadly. Along the way, we
hope to observe whenever possible how our foundation of “good number sense” dictates
how we proceed with symbols. Try to stay with it!
8
A term is either a single number or variable, or the product of several numbers and/or
variables.
For example, in 3 + 4x + 5yzw 3, 4x, and 5yzw are all terms.
The word “term” is from the Latin terminus: “boundary line, limit,” and from the Indo-
European word root ter-: “peg, post, boundary.”
In the above example, 4x is “bounded” by the two plus signs, while the other terms have
those plus signs and the ends of the expression as boundaries.
The “numerical part” of a term (such as the 4 in 4x) is called a coefficient.
Similar terms are terms that contain the same variables raised to the same powers.
Only similar terms can be added/subtracted.
To add/subtract similar terms:
Add/subtract the coefficients and copy the variables – do not change them.
So, for example, xxxx 592814 … because 592814 !!!
But, in contrast, 242433423342 171152 wvtwvtwvtwvt is done – already simplified.
Tips:
▪ Copy the terms carefully and be careful when combining!
Terms like yzx 2 and zxy 2 look a lot alike, but they aren't similar and cannot be combined.
▪ Don't overlook terms that are alike!
Terms obey the commutative property of multiplication - that is, xy and yx are similar
terms, as are:
725 zyx and 275 yzx and 572 xzy and 752 zxy and 527 xyz and 257 yxz
▪ Group and combine similar terms (with signs correctly interpreted) freely!
The commutative and associative properties allow it!
9
Did you know there’s some “work” missing in a previous example?
xxxxxx 559281492814
For a mere instant, the x “factors out” of the expression, leaving the numbers to “add up.”
xxxx 9281492814 This is an example of the distributive law.
I never write that step – and you probably won’t either, once you become accustomed to the
process. Yet the distributive law is a major key to simplifying expressions.
“Opposite”: If E is an expression, then –E is called the opposite of E.
► To find –E, take the opposite of every term of E.
Examples:
i. 11741174 yxyx ii. 53325332 109109 zxyxzxyx
This “opposite” idea can assist distributions involving negative factors:
“Multiply by the absolute value of the number, and switch all the signs”
Example:
7842543060784254306013795106 zyxwzyxwzyxw
First the 6 was distributed to all terms, then the signs were switched.
“Nested”: A nested expression is an expression bounded by grouping symbols that is part
of a larger expression. Certain applications yield nested expressions, so it is necessary to
become comfortable with their simplification. The challenge is to not be intimidated by the
appearance and to closely follow the Order of Operations.
Example: xx 32553958 Solution:
87858785
15457240
51537240
1510537240
32553958
xx
xx
xx
xx
xx
10
Exercises: Simplify.
a. 5m – 12m b. 3c – 5 – c c. 5 – h + 2
d. r – 7 + 2r – 3r e. 22 426 aaaa f. p + 2q – 4q + 5p
g. 22326 www h. 6r + 4s – 3r – s i. 10r – 6t – 2 + 2t – 4r
j. bb 4110 k. y 949 l. r3621
m. 6525 m n. 14344 pp o. 324 xx
p. 54671467 2323 xxxxxx q. 17645653 2324 xxxxxx
r. 1641765 3234 xxxxxx s. 2424 3361637 xxxxxx
t. 346543 22322 xyyxyxyyx u. 4324 72253112 mmmmmm
Solutions:
a. –7m b. 2c – 5 c. 7 – h d. –7
e. aa 103 2 f. 6p – 2q g. 1292 2 ww h. 3r + 3s
i. 6r – 4t – 2 j. 10 – 6b k. 4y – 27 l. – 6r – 11
m. 10m + 19 n. 20p + 8 o. 3x + 11 p. 612 2 x
q. 443 234 xxxx r. xxxx 5725 234 s. 1433 24 xxx
t. 7559 322 yxyyx u. 1524 234 mmm
11
An equation is a statement that two expressions are the same.
This “sameness” is written with the equality symbol, =
An equation that is always true is called an identity.
Examples: 2 + 3 = 5 and x − x = 0
The “opposite” of an identity, an equation that is always false, is called a contradiction.
Examples: 2 + 3 = 1 and x + 1 = x
The following equation is neither an identity nor a contradiction: x + 1 = 2
It is called a conditional equation because it is only true under certain conditions (in this
case, if x = 1).
The values of the variables for which an equation is true are called solutions.
To solve an equation means to find its solutions.
Given any equation, the following operations may be used to produce another true equation:
Any quantity can be added to both sides.
Any quantity can be subtracted from both sides.
Any quantity can be multiplied to both sides.
Any nonzero quantity can divide both sides.
For now, we are going to focus on very basic equations: one variable only, and no powers of
that variable or division by that variable. Such equations are called linear equations of one
variable.
We will learn to use the operations skillfully enough to consistently produce correct
solution(s)!
Many have described the overall process as isolating the variable.
An equation with the variable completely isolated could be called a ”solution equation.”
12
Examples:
Solve.
1. 2580 t 2. 12111 w 3. 7.1021.9 p
4. c5
6
15
16 5. 14166 q 6. 51913 ss
Let’s look at the solutions carefully.
1. 2580 t
Solution: If 80 is added to both expressions, we achieve a solution equation:
105
1058080
80258080
2580
t
t
t
t
2. 12111 w
Solution: This time, dividing each expression by –11 yields the solution equation:
11
1111
1
1
11
111211111
12111
w
w
w
w
3. 7.1021.9 p
Solution: Add 10.7 to both expressions, then divide both by 2:
p
p
p
p
8.0
2226.1
26.1
7.107.1027.101.9
Note: the variable may be
isolated on either side – no
difference in solution!
13
4. c5
6
15
16
Solution: In example 2, I used the reciprocal to show how –11 “cancels out,” leaving w
isolated. Here, I use that idea and multiply each expression by 6
5 :
c
c
c
c
98
5
6
15
16
65
56
9080
65
56
65
1516
5. 14166 q
Solution: Add 16 to both expressions, then divide both by 6:
5
63066
306
161416166
q
q
q
q
6. 51913 ss
Solution: First, we separate the
variables from the numbers. This can be
done two ways.
Option 1: Add 19 to both expressions
and subtract s from both:
1412
1413
1413
195191913
s
ssss
ss
ss
I can divide both expressions by 12 to
finish:
67
12141212
s
s
Option 2: Add 5 to both expressions and
subtract 13s from both:
s
ssss
ss
ss
1214
13131413
1413
5551913
I can divide both expressions by –12 to
finish:
s
s
67
12121214
14
The steps that I used are so systematic that I can list them for you:
An Equation-Solving Procedure
1. Add or subtract on both sides to get all variable terms on one side and all constant
terms on the other side – separate the types of terms.
2. Multiply or divide by the remaining variable coefficient to isolate the variable.
3. Check all possible solutions in the original equation – evaluate for consistency.
The only way to make any procedure meaningful for you is to practice – as you probably
know. Find practice opportunities in class, online and in text resources.
Before leaving the topic of solving entirely, let’s consider a couple more examples.
Examples:
7. xx2
5
10
7
4
3
5
2
Solution:
Here’s a very direct approach, with a bit of fraction arithmetic:
2
1
29
10
20
29
10
29
29
10
20
29
10
29
4
3
10
7
4
3
4
3
10
29
10
7
4
3
10
29
2
5
2
5
10
7
2
5
4
3
5
2
2
5
10
7
4
3
5
2
x
x
x
x
x
xxxx
xx
2
1
12
11
29
10
20
29
20
20
29
20
15
20
14
54
53
210
27
4
3
10
7
10
10
29
10
25
10
4
52
55
25
22
2
5
5
2
1
1
2
1
LCD
LCD
Arithmetic steps:
15
It’s not a bad process, but I’ve got another way to proceed…
Consider the Least Common Denominator for all terms initially:
xx2
5
10
7
4
3
5
2 LCD = 20
Convert each term to that denominator:
xxxx20
50
20
14
20
15
20
8
102
105
210
27
54
53
45
42
Now make each side of the equation 20 times larger by “clearing” denominators:
8x + 15 = –14 – 50x
Now, solve: 8x + 15 = –14 – 50x
+50x +50x
58x + 15 = –14
–15 –15
58x = –29
÷58 ÷58 2
1 x
It’s the same solution, since it was truly an equivalent equation. Each side was 20 times larger.
So this could be a much better way to go if you “hate fractions”…
8. 0.04 – 0.11x = 0.4x – 0.98
Solution:
Again, it’s great if you see no difficulty here: 0.04 – 0.11x = 0.4x – 0.98
+0.11x +0.11x
0.04 = 0.51x – 0.98
+0.98 +0.98
1.02 = 0.51x
÷0.51 ÷0.51 x 2
16
However, since the coefficients are decimals, also called decimal fractions, they are also
subject to “clearing.”
In order to do so, one must achieve equal place value for all terms:
0.04 – 0.11x = 0.40x – 0.98 All are hundredths.
Now make each side 100 times larger by “clearing” the decimal points:
4 – 11x = 40x – 98
Now observe that the solution of the resulting equation is also x = 2:
4 – 11x = 40x – 98 4 = 51x – 98 102 = 51x
+11x +11x +98 +98 ÷51 ÷51
2 = x 2 x
Admittedly, this may not seem so exciting as clearing fractions from the previous example,
but it is equally valid and available should you choose to employ it.
With clearing in mind, we present an “enhanced” sequence of steps for the method:
An Equation-Solving Procedure
1. If necessary, simplify each side first: distribute, combine similar terms, etc.
2. If desired, clear any fractions or decimals.*
* - Clearing…
…Fractions:
a. Then, find a common denominator for
all terms involved in the equation.
Convert all terms to equivalent fractions,
with that common denominator.
b. Clear the terms of the denominator.
This will leave the numerators behind,
which are still equivalent as “parts.”
…Decimals:
a. Give all terms equal place number. This
will often involve place-holder zeros.
b. Clear the terms of decimal point. This
will leave whole numbers behind, which are
still equivalent in “relative” value – as parts.
17
3. Add or subtract on both sides to get all variable terms on the opposite side of the
constants – separate the types of terms.
4. Divide or multiply by the simplified variable coefficient to isolate the variable.
5. Check all possible solutions in the original equation – evaluate for consistency.
Exercises:
Solve.
1. 33
20
2
1
3
2
2
1 xx 2.
56
29
2
1
7
1 x
3. 200
151
8
7
5
3 x 4. 003.05.00022.014.0 x
5. 4.03.05.05.2 x 6. 1400005.03.002.0 y
Solutions:
1. 63.0117 x 2. 75.04
3 x 3. 2.051 x
4. 272787.127500,5033,700 x 5. x = –18 6. 35.333,23320
667,666,4 x
The correct decimal forms are given for your convenience…
18
Question: What would change if, instead of equality, we are given inequality?
This must be considered to complete the trichotomy of real number comparison.
For any two real numbers a and b, exactly one of the following three statements is true:
a = b a > b a < b
a equals b or a is greater than b or a is less than b
Here you can see the three symbols for relations of order: ,,
You might recall two other symbols used in the analysis of inequality: ≥ and ≤
These are “blends” of the first three: means or , while means or
Answer: “Not much”…
♦ An even more “enhanced” sequence of steps for the method:
A Procedure for Solving a Linear Equation or Inequality
(Expression) (relation of order)_ (Expression)
1. If necessary, simplify each side first: distribute, combine similar terms, etc.
2. If desired, clear any fractions or decimals.
3. Add or subtract on both sides to get all variable terms on the opposite side of the
constants – separate the types of terms.
4. Divide or multiply by the simplified variable coefficient to isolate the variable.
So far, nothing different between equations and inequalities.
It’s the next step that introduces the distinction.
We will cover all three cases: ,,
19
5. a. = stays = equal stays equal
b. > and < The symbol will stay the same unless both sides are divided or multiplied
by a negative number – in that case, the symbol will “flip”
For example: 366 x cannot be true unless 6x (right?)
Here’s how the “steps” might look if you try to solve: 636
66366
xx
In for the end result to be logically consistent, we must reverse the relation: 6x
A verbose way to explain it might be,
“Making both sides of an inequality into opposites creates the opposite inequality.”
The nature of an inequality also demands a bit more explanation when it comes to writing
down a solution.
That is, saying something like, “ x > 3 ” is actually kind of vague…do we mean one number, as
long as it’s more than 3? Or a couple of numbers, both more than 3? A few numbers?
Several?
NO... we mean all numbers larger than 3 when we write x > 3.
So, now that we have clarified the issue, should we just assume that everyone else in the
world understands it as we do?
NO… instead, we illustrate the situation with a number line graph.
That’s what it will look like! There is a boundary value, emphasized by a dot above it, and a
ray (a “half-line”) signifying numbers larger or smaller than that boundary.
Notice how when we have ≤ or ≥, the dot is solid, because of “or equal to.”
In contrast, when we have < or >, the dot is empty, for equaling the boundary value isn’t
allowed.
x ≥ A
x > A
x ≤ A
x < A
A
20
You may want to consider the following “rule of thumb:”
If the variable is on the left side, the graph points the same way as the inequality symbol.
This leads us to the final “step” of our solving process.
6. Check your solution in the original statement – evaluate for consistency.
For an equation, this means to use the value which you’ve isolated for the variable in the
original equation to verify that the equation is, in fact, true for that value.
For an inequality, the situation is a bit more intimidating – we are finding infinitely many
solutions, after all! However, checking may simply entail using a single value from the
solution set in the original inequality to make sure that the inequality is true for the entire
solution set.
For example, if the result of your isolation claims that x < 10, then using x = 9, x = 8 or even x
= 0 would be a “fair” selection and should be enough to convince oneself that the correct set
of solutions has been found.
Here are some examples of solving inequalities and graphing the solution.
Examples:
1. 4x – 9 > 39 – 12x 2. 11.5 – 2.1x < – 0.9x – 29
3. 8x – 27 ≤ –10x – 29 4. xx6
21
3
8
8
7
2
3
The solutions are shown on the next page…maybe you could try them first.
21
Solutions:
1. xx 123994
+12x +12x
39916 x
+9 +9
4816 x
÷16 ÷16
3x
Nothing notable so far…?
2. 11.5 – 2.1x < – 0.9x – 29
115 – 21x < – 9x – 290
+9x +9x
115 – 12x < – 290
–115 –115
–12x < – 405
÷(–12) ÷ (–12)
x > 75.334
13512
405
Decimals were cleared first.
Division by an opposite (negative 12) forces the
opposite relation…in this case, “greater than”
instead of “less than.”
3. 8x – 27 ≤ –10x – 109
+10x +10x
18x – 27 ≤ –109
+27 +27
18x ≤ – 82
÷18 ÷18
x ≤ 5.49
4118
82
Division of an opposite (negative 82) does not force
any change in the relation.
4. xx6
21
3
8
8
7
2
3
xx24
84
24
64
24
21
24
36
– 36x + 21 ≥ 64 + 84x
+36x +36x
21 ≥ 64 + 120x
–64 –64
– 43 ≥ 120x
÷120 ÷120
12043
12043
x
x
Fractions were cleared first.
0
0 33.75
0
0 3
22
A Procedure for Solving a Linear Equation or Inequality
1. If necessary, simplify each side first: distribute, combine similar terms, etc.
2. If desired, clear any fractions or decimals.*
* - Clearing…
…Fractions:
a. Then, find a common denominator for
all terms involved in the equation.
Convert all terms to equivalent fractions,
with that common denominator.
b. Clear the terms of the denominator. This
will leave the numerators behind, which are
still equivalent as “parts.”
…Decimals:
a. Give all terms equal place number. This
will often involve place-holder zeros.
b. Clear the terms of decimal point. This
will leave whole numbers behind, which are
still equivalent in “relative” value – as parts.
3. Add or subtract on both sides to get all variable terms on the opposite side of the constants
– separate the types of terms.
4. Divide or multiply by the simplified variable coefficient to isolate the variable.
5. a. = stays = equal stays equal
b. > and < the symbol will stay the same unless both sides are divided or multiplied by
a negative number – in that case, the symbol will “flip”
6. Check your solution in the original statement – evaluate for consistency.
Exercises:
1. 3
82
5nn
2. 42
64
3
mm 3. vv 13523
4. 27118 yy 5. 134315 xx 6. ww 216452
7. zz4
1
8
3
8
7
23
Solutions:
1. 518n
2. 8m
3. 4v
4. 13y
5. 4x
6. 3w
7. 1z
0 1
0 3
0 4
0 13
0 4
0 8
3.6 0
24
Now we begin the study of how quantities are represented in expressions. If we can combine
these expressions in equations, we will be able to solve for the quantity we desire.
Expressions are often very literal; the words used may have direct translation to variables and
operations. So it really comes down to knowing how our operations and groupings are spoken.
Examples:
1. The difference of five times the sum of a number and 13 and 40
Solution: 40135 N
Note that this can be simplified as follows: 2554065540135 NNN
2. 50 more than half of the difference of the number and 200
Solution: 2002
150 N
This can be simplified as follows: 502
11002
1502002
150 NNN
Here are some popular “math phrases,” grouped according to their respective operation.
Addition
plus
increased by
more than, greater than
combined, together
total of, sum of
added to
Subtraction
decreased by
minus, less
difference between/of
less than, fewer than
reduce by
Multiplication
of
times, multiplied by
product of
double, triple, etc.
twice, thrice
Division
divided by
ratio of, quotient of
percent (divide by 100)
per, a, for each (?)
Equality
is, are, was, were, will be, would be, should be
gives, yields
sold for
25
Do you think this is an “unrealistic” exercise? Do you believe that mathematical phrases don’t
happen anywhere but in a math classroom?
Observe, if you will, these phrases lifted from news articles found on the internet…
…is on track to double output…
2x
…200 more than last year's target…
200 + x
…when I raised the amount of sugar again by half…
x + 0.5x
…$20 more than twice last night’s receipts…
20 + 2x
… reduce your caloric intake by 500 calories per day…
x – 500
…combined wages, tips and salaries…
W + T + S
…for $4,000 less than last year’s total…
x – 4,000
…50 percent thinner than it was…
0.50x
…nine-tenths of the rate of the competitor…
0.9x
…there will be 4 stickers per child…
4x
…twice length and width gives a rectangle’s perimeter…
2(l + w)
…the grade point average is calculated by multiplying the quantitative [grade] values by the credit
value of the correlative course, and then dividing the total by the sum of all credits…
n
nn
ccc
cgcgcg
21
2211
26
Finally, continuing the theme of translation, here are some phrases that correspond to
inequalities.
Most of these will typically be preceded by some conjugation of the verb TO BE, like IS.
> <
Greater than Less than Greater than or equal to Less than or equal to
More than Below At least At most
Exceeds Under Cannot be less than Cannot be greater than
Above Exceeds Does not exceed
No less than No more than
We may not have compiled a complete list for expressions, equality or inequality, but it isn’t a
bad start!
The Bad News…
Even when you gain skill in translating expressions from words to symbols, it gets tougher still –
for expressions surely arise from applications, usually called word problems.
But there is a sequence of steps that is known to be effective in systematizing, and hopefully
simplifying, the process…
Mathematical Modeling
1. First you must comprehend the question.
What is being asked? What quantity will be the final answer? Would a table of information be
useful to organize the given information? Would it help to draw a picture of the context?
2. Then you must translate the question to expressions, as well as an equation or inequality.
How many quantities are unknown? Should they each have their own variable? Or are they
related quantities, so that perhaps the same variable can be involved in representing both?
What are the key words? Where is equality or inequality expressed?
3. Then you must solve the equation or inequality that you found.
Surprisingly, this may be comparatively simple, measured against the first two steps.
27
4. Then you must answer the question, and possibly check your answer as well.
Sometimes checking your answer may mean doing a “reality check:” Can that number really be
the solution? Also, remember that to answer correctly, one must include units (if any) in the
solution. Also, don’t forget to answer the correct question: for example, if a question asks for
area, don’t give the perimeter instead!!!
Example:
Vince is fencing some of his property, and plans to make a rectangle with length 30 feet more
than the width. He buys 1,140 feet of fencing to surround the perimeter of the field.
What is the area of his newly fenced field?
Solution: 1. Comprehend: Here is a drawing of the situation:
2. Translate: I will label the diagram with expressions that represent the given information:
“…length 30 feet more than width…” “…perimeter…”
An equation can then be formed from the statement “1,140 feet of fencing…the perimeter”:
140,1460 w
3. Solve: Solving this equation is pretty decent…
field
length
width
I recall that perimeter is twice length and width…
w
ww
ww
wlwl
460
2260
2302
222
It doesn’t matter which sides you label as length and width
as long as you don’t do so with opposing sides, which are
assumed to be equal for any rectangle.
Three P’s of Advice: be Patient, Precise, and Persistent.
270
4080,1
080,14
60140,14
140,1460
w
w
w
w
w
field
length = l = 30 + width = 30 + w
width = w
28
I now know the width to be 270 ft, and I also know that the length must be 30 ft more, which
is 300 ft.
4. Answer: The question asked for area, which is the product of length and width.
300270 = 81,000 square feet
Common types of Mathematical Modeling in Beginning Algebra Courses:
Number puzzles – “A number is…”: These will range from very simple to very
challenging…and they are great for practicing your Modeling process, since the
statements will typically be able to translate very directly.
Geometry-based questions: Perimeters and area of triangles, rectangles, et cetera;
angles within a triangle; proportion or other length ratio questions.
Questions based on distances, (constant) speeds and times: These can be very
tough. The key is remembering that distance = rate × time, which is usually symbolized
as d = rt. A popular way to comprehend the information in this type of question is to
make a table to organize distances, rates and times:
Questions based on constant rates: Steady rain, hourly salary, vehicle rental, questions
of age (which increases 1 per year)…these are similar to the distance questions, and may
translate similarly, since in both types the rate is constant. That’s a pretty big deal,
actually…
Further Examples:
a. Jerry is traveling around the country. So far, he has traveled 6,784 miles, 9
8 of his total trip.
How many more miles does he have to travel?
Solution: A very direct approach is to let T represent the total length of his trip, in miles.
Then we can translate the given information, again very directly, as follows:
9
8 of his total trip is 6,784 miles → 784,698 T
Distance = Rate × Time
29
This equation can be solved in one step, by multiplying both sides by the reciprocal of 9
8 :
632,7784,6784,689
98
89
98 TT
So T = 7,632, and therefore the remaining distance is 7,632 – 6,784 = 848 miles.
b. Consider the following two numbers: one is 20 more than the other. If each number were
increased by eight, the larger number would be exactly three times the lesser number.
What are the numbers?
Solution: This contains, again, the possibility of a very direct approach:
Let N represent one of the numbers, and then the other must be 20 + N to be 20 more than
the first.
To increase each number by eight, we simply add: N + 8 and 20 + N + 8 → N + 28
Now we reach the “big” statement of equality:
…the larger number would be exactly three times the lesser number → 8328 NN
Notice that N + 28 has to be the larger number…it is, after all, still 20 more than the other.
Solving this equation is, by now, hopefully very routine:
N
N
NN
NN
2
24
24328
8328
This means the lesser number is 2, so the larger is 20 + 2 = 22
c. The sum of three consecutive odd numbers is 81.
What are the numbers?
Solution: It’s a theme – a direct approach can be successful here.
848 miles
2 and 22
30
Consecutive triplets of odd numbers such as 3, 5, 7 or 29, 31, 33 clearly have separations of
two.
So if we say the first unknown odd number is N, then the next two are N + 2 and N + 2 + 2
So we have N, N + 2 and N + 2 + 2 → N + 4
The sum is 81, so we must now state:
N + N + 2 + N + 4 = 81 This is quite nice, and solves quickly:
25
753
8163
8142
N
N
N
NNN
The first number is 25, next is 25 + 2 = 27, and next is 25 + 4 = 29
d. Bob wants to have an average score of at least 180 for the three games he bowls in the
league tournament. He bowls 166 and 189 in his first two games.
What must he bowl in the third game to achieve his goal?
Solution: One last time, why not try a direct approach?
Let G represent his score in the third game. The average of the three games is found by taking
their sum and dividing by three.
Therefore an expression for the average is: 3
355
3
189166 GG
Now for the final translation: Bob wants to average at least 180. This is an inequality, and
synonymous with “greater than or equal to.”
Thus we have: 1803
355
G We can solve by clearing the denominator and then isolating G:
185
540355
18033
3553
G
G
G
A score of at least 185 will satisfy Bob.
25, 27 and 29
He must bowl at least 185 in game 3
31
Exercises:
a. Bob takes a taxi from the airport to his home. It charges $2.20 for the first mile, and 35
cents for each additional mile. He is charged a fare of $15.50 for his ride.
How many miles does he live from the airport?
b. Ron is fencing a garden, and plans to make a rectangle with length 10 feet less than thrice
the width. He buys 380 feet of fencing to surround the perimeter of the garden.
If each new plant requires 3.5 square feet of space, how many can he plant?
c. Phil gave one third of his daily apple harvest to Tom, one quarter to Keith and Donna, one
sixth to Bruce, one eighth to Billy and one twelfth to Mick. He had one apple left for himself.
How many apples had he harvested that day?
d. The width of a rectangle is 16 yards.
For what lengths will the area exceed 264 square yards?
e. 6 times a number is greater than the sum of the number and 30.
What number(s)?
Solutions:
a. 39 miles b. 2,000 plants c. 24 apples d. Exceeding 16.5 yds e. Greater than 6
32
It might be worth noting that many students find this category of questions frustrating –
some because they may be difficult to solve, and some because they may be difficult to solve
algebraically.
There are those who find word problems so difficult that they will refuse to engage in any
attempt.
There are also those who find it insulting to have their own creative approach stifled by having
to solve using a variable and an equation, so they will also refuse to engage in any attempt.
My advice is, simply, not to refuse – but to try.
33
Percentage is a very common way of expressing parts of a whole and changes in quantity,
and it deserves particular attention as a Modeling issue.
The symbol % signals that a number is expressing a percentage; it also reminds us that the
number must be converted before performing any computations!
A convenient tool for conversion of percentages into decimal numbers comes from the
definition;
Percent: “Out of one hundred” Thus 42077.0100
077.42%077.42
One must move the decimal point two places to the left in order to remove the % symbol.
This also means that to use the % symbol, one must move the decimal point two places to the
right.
Alphabetical order of the first letters of “Decimal” and “Percentage” mirrors the decimal point
shift. Right?
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
ALL percentage questions can be resolved using the trichotomy which follows. Some may take
more time to decipher and “set up” according to one of these types, but that’s the way it goes.
TYPE I
■ Question: What number is P percent of A? Equation: PAx
Example: My dental insurance paid 70% toward the $780 bill for my latest surgery. How much
did they pay?
Question: What number is 70% of 780? Equation: x = 0.70 · 780
Solution: x = 546 ► They paid $546.
D to P…move to the right
P to D…move to the left
34
TYPE II
■ Question: A is P percent of what number? Equation: PxA
Example: My beverage label says that 25 mg of sodium is 1% of Recommended Daily
Allowance. What must that Recommended Daily Allowance be?
Question: 25 is 1% of what number? Equation: 25 = 0.01x
Solution: 500,201.0
25 x ► Recommended Daily Allowance must be 2,500 mg.
TYPE III
■ Question: A is what percent of B? Equation: BxAxBA
Example: 2 days of each week are designated as the “weekend.” What percent is that?
Question: 2 is what percent of 7? Equation: 2 = 7x
Solution: 285714.07
2 x ► The weekend is 571428.2828 74 % of a week.
Notice that in Type III, the answer is a percentage that must be converted before using the %
symbol. According an earlier discussion, move the decimal point 2 places right, and done!
Another (correct) alternative to convert to percentage is to multiply by 100%; this “makes sense”
because 100% = 1, numerically speaking.
Taking the example of Type III (above): %7
428%
7
200%100
7
2
7
2x
Notice that the number 7
2x was actually multiplied by 100, and the % symbol “sticks.” That
also makes sense: the literal meaning of % is the phrase “out of 100.”
To find how many parts that is, multiply by the 100 to solve it…
%100%100100
% xxxxxx
This means that every number is 100 times larger when viewed as a percentage.
35
Exercises:
1. What number is 35% of 200?
Solutions: 70
2. 6 is 15% of what number?
40
3. 30 is what percent of 20?
150%
4. What number is 102% of 2,000?
2,040
5. What percent of 80 is 60?
75%
6. 14 is 70% of what number?
20
7. What number is 0.5% of 3.2?
0.016
8. 2.5 is what percent of 4?
62.5%
9. 5 is what percent of 15?
33⅓%
10. 12.5% of 32 is what number?
4
11. What percent of 8.7 is 17.4?
200%
12. What number is 3.1% of 60?
1.86
36
Next we consider formulas. A formula is an equation of two or more variables. That’s all.
So, we can do what we do with equations we’ve already encountered – solve them.
To solve for a variable, use the normal steps to isolate that variable.
While solving for any one variable, treat all other variables as constant.
Example: Let CBA 642 , and solve for A.
Solution: CBA 642 Isolate it on the left side
BCA 462 Subtracting 4B from both sides
2
46 BCA
Dividing both sides by 2
BCA 23 Reducing to lowest terms
So BCA 23 is an equivalent formula, but solved differently.
Example: Let SR
V11
, and solve for R.
Solution: SR
V11
Do there have to be fractions?
RSRSV Clearing fractions with common denominator RS
SRRSV Subtracting R from both sides
SSVR 1 Factoring R from those terms on the left side
Factoring is kind of like “un-multiplying” or “anti-distributing.” It is writing an expression in the form of a product.
1
SV
SR Dividing both sides by SV – 1
Example: If 4x – 6y = 20, solve for y.
37
Solution: 2064 yx Isolate it on the left side
xy 4206 Subtracting 4x from both sides
6
420
xy Dividing both sides by –6
xy3
2
3
10 Reducing both sides to lowest terms
Example: If
V
AdVd
50, solve for Ω.
Solution:
V
AdVd
50 Have to get it out of the denominator
AdVV d 50 Multiplying both sides by V50
dVV
Ad
50
Dividing both sides by dVV 50
Consider the formula 1 xy .
For a chosen value of x (or y), there will always be a corresponding value of y (or x).
For example, if one chooses x to be 3, then y must be 4, because 4 – 3 = 1.
If that doesn’t arrive so quickly to your mind, you can find y by using x = 3 in the formula!
4
13
1
y
y
xy
This pair of numbers, together, forms one solution to the formula.
Typically, the solution pair is written in this fashion: (3,4)
Since the numbers have a specific order (x, then y), we call (3,4) an ordered pair.
Now here’s where it gets interesting. First, there is no end to the supply of ordered pair
solutions for this formula. Think about it: how many numbers can be chosen to be x (or y)?
38
There are infinitely many to choose, so there are infinitely many ordered pair solutions. For
example, you should be able to verify that both (0,1) and (– 4,–3) are solutions.
Next, an amazing thing happens when we display these ordered pair solutions using the
Cartesian plane. The Cartesian plane is a pair of number lines set perpendicular to each other,
like this:
The convention is to let the horizontal (left-to-right) number line represent the x-value, and the
vertical (up-and-down) number line represent the y-value. Then an ordered pair can be plotted
as a point in the plane, as follows:
Practice this by plotting the pairs (0,1) and (– 4,–3).
Notice anything about these three solutions to the single equation y – x = 1?
They “line up” very neatly.
That’s a convincing reason why these types of equations are called linear.
A linear equation in two variables has form CByAx , with at least A or B nonzero.
The solution to such an equation is graphed as a straight line consisting of the infinitely many
ordered pairs that actually solve it.
The “middle” is always assumed to be 0 on
both lines, and is called the origin.
The origin
(3,4) (1,0)
(– 4, –3)
3
4 (3,4)
Move 3 units to the right, and then
move 4 units up, and plot a point.
39
There are several techniques for graphing such an equation. Here is a relatively simple one.
The Intercept Method:
1. Complete the ordered pairs 0_, and _,0 and plot them.
0_, is the x-intercept (with y = 0) and _,0 is the y-intercept (with x = 0)
2. Connect with a line.
3. (optional) Check a third ordered pair against the equation to ensure the linearity.
Example: 2085 yx
Solution: 0_, _,0
–5x = 20 8y = 20
x = – 4 y = 2.5
0,4 5.2,0
What’s next?
The slope of a line in the Cartesian plane is generally represented by the letter m, and is defined
as the change in the y-coordinate divided by the corresponding change in the x-coordinate,
between two distinct points on the line.
This quantity deserves to be represented symbolically and explored.
Given two points 11 , yx and 22 , yx , the change in x from one to the other is 12 xx , while
the change in y is 12 yy .
Substituting both quantities into the equation above, one obtains the slope formula:
Slope = x
y
xx
yym
12
12
The Greek capital letter delta ( ) is commonly used in the sciences to mean “difference in” or
“change in.”
(2.5,0)
(– 4,0)
40
Since the y-axis is (conventionally) vertical and the x-axis is horizontal, the slope formula is often
memorized as “rise over run”, where y (read “delta y”) is the “rise” and x (“delta x”) is the
“run.”
A second way to view it could be “rise over right,” as x is always measured in the positive
direction.
Therefore – again – m is equal to the change in y, the vertical coordinate, divided by the change
in x, the horizontal coordinate; that is, m is the ratio of the changes, usually simply called the
rate of change.
Note: You may observe that using the Intercept Method for graphing lines provides an
excellent diagram from which to “see” the slope, even compute it mentally…
…but with a little work, one can see the slope revealed as the coefficient of x when y is solved.
That is, in the formula bmxy , m represents the slope of the line. This is very significant!
What is b? Well, note that if x = 0, then y = 0x + b = b, so there is an ordered pair b,0
This is the y-intercept, so we can identify b as that point if we wish.
For all of these reasons, bmxy is called Slope-Intercept Form of a linear equation in two
variables.
Example: Graph the solutions of the linear equation 30155 xy
Solution: You should verify that the intercepts are 0,2 and 6,0 , so the graph is as follows:
: “run”
: “rise”
6
–2
y
x
41
As mentioned, the intercepts make a great representation of the slope. Moving left-to-right
from the x-intercept to the y-intercept, we can observe a rise of 6 and a run of 2.
This means we have slope = m = 32
6
Now let’s solve the equation for y: 30155 xy
+15x +15x
555
30155 xy
y = 3x + 6
Check it out – when the equation has y isolated, slope and y-value of the y-intercept are plain to
see!
It becomes especially neat to use this information for sketching the graph. It’s quite direct,
really – start at the y-intercept and follow the slope…
So we see that Slope-Intercept Form reveals important information and serves as a tool for
graphing.
At this point we’ve found two forms of a linear equation with two variables, each with its own
use(s)/technique(s)/information. Are there others? Yes, and we pause to summarize them here:
Forms of a Linear Equation with Two Variables
■ General Form: CByAx A, B and C constant
Mostly used for identification of the type of object with which one is working.
slope y-intercept
value
6
By moving up 3, right 1, we
find ourselves at another point
on the line…
…find that point by adding
coӧrdinates:
9,136,10
42
■ Slope-Intercept Form: bmxy m = slope, y-intercept b,0
Extremely useful for graphing and application, it reveals perhaps the two most vital pieces of
information about a line.
■ Point-Slope Form: axmby m = slope, ba, any point on line
This comes from the slope formula, really. If yx , is a generalized point on a line and ba, is a
specified point also on that line, then the slope between the points is ax
bym
The slope formula, therefore, can be re-interpreted as an equation of the line itself.
■ Intercept Form: 1d
y
c
x x-intercept 0,c , y-intercept d,0
Surprisingly under-used, this form can be directly applied when both intercepts are known. That
could be useful from time to time.
The last topic of linear equations with two variables is linear modeling. Because it is a very
fundamental skill for interpreting real-world data, we review it briefly here.
Linear modeling: Using a line to connect data, forming a linear equation with two variables.
Often that equation is used to extrapolate (outside the data points) or
interpolate (between the data points) new information.
Question: How can I turn two points into a linear equation?
Dave says: “Well, unless you’re given the intercepts…”
There are several ways to do it. My favorite – the method I use more often than any other – is
using the point-slope formula.
Example: Find an equation of the line joining 3,4 and 2,2
1 .
Solution: First, find the slope:
910
9
2
1
5
29
5
42
1
32
x
ym
Substitute the slope and one of the points into the formula: axmby
43
Here, that would result in either of these two options:
43
43
3,4
910
910
xy
xy
21
910
21
910
21
2
2
2,
xy
xy
Although it may not look so, these equations are equivalent. One way to see it is by solving
for y to gain slope-intercept form…
913
910
940
910
940
910
3
3
xy
xy
xy
913
910
95
910
95
910
2
2
xy
xy
xy
Same eventual equation.
You can get to slope-intercept form directly by memorizing it: bmxy
Next use your computed value of slope m with a point yx , to solve for b:
b
b
b
913
940
910
3
43
3,4
b
b
b
913
95
21
910
21
2
2
2,
Both point back to the already-named result, 913
910 xy
Slope is key – computed between two points, the equation can be based on one point.
Question: What if it’s a word problem?!?
Dave says: It still works from the slope. You can proceed very formally (creating an
equation to use), or informally (using slope in a proportion).
Formal approach
Step One: Name the data points as ordered pairs 11 , yx and 22 , yx .
44
Reminder: The little 1’s and 2’s are subscripts; they are simply labels to distinguish the 2 points.
1x is read “x sub-one.”
One must choose wisely: as a general rule of thumb, y must depend on x!!!
Step Two: Compute the slope; as you know, many “formulas” are memorized for this:
Slope = run
risex
y
xx
yy
xx
yym
21
21
12
12
Note: Slope is a rate, so it always has units in linear models; that is, it’s always “something
per something.” See below.
Step Three: Derive a linear equation form:
Slope-Intercept Form
y = mx + b
m = slope
b = y-coördinate of y-intercept
If b is unknown, it must be found by
substitution of one of the two data
points and the slope.
Point-Slope Form
00
00
mxymxy
xxmyy
m = slope
00 , yx = any point on the line
Step Four: Use the equation to find the new information. This means identifying an
incomplete ordered pair; that pair containing a given fact that leads to the
(unknown) answer.
Note: This will involve substitution of a given value of y (or x) and solving to find an unknown
x (or y)
A “classic” scenario:
Example:
Suppose that a certain strain of pea plant requires 14 days to reach a height of 6 inches and 30
days to reach a height of 16 inches. When will it reach 6 feet?
You might try estimating an answer before you look at what I’ve got next…
45
Solution:
1. Since height depends on time (at least, that’s the conventional view), the ordered pairs I will
use are 6,14 and 16,30
Be cautious here; simply copying information in “the order it’s given,” as I did here, can be
hazardous.
If the question read, “…and reaches 16 inches after 30 days,” would you still have the right pairs?
2. This yields a slope of 8
5days8
inches5
days16
inches10
days1430
inches616
m -inch per day
3. Using Point-Slope Form, I can immediately state the following using the point 6,14 :
4
11
8
5
4
35
4
24
8
5
4
356
8
5
8
706
8
514
8
56
8
5
xy
xyxyxyxy
The point 16,30 could also be used to gain this equation. As long as the slope is correct, it will
work out fine, assuming no other mistakes are made!
Alternatively, I can use Slope-Intercept Form with the computed slope and one of the two data
points.
bxy 8
5
Using 6,14 :
bbbb 4
11
4
35
4
24
4
35614
8
56
Hence 4
11
8
5 xy , as before. You should check that using 16,30 gives the same result.
4. Finding when the height will be 6 feet is the same as completing the ordered pair 72,x
Here’s how: First set 4
11
8
572 x , as we have y = 72.
Then we can solve for x.
46
days6.1195
598
1
299
5
2
8
5
5
8
4
299
5
8
8
5
4
299
8
5
4
11
4
288
8
5
4
1172
x
xx
xxx
Informal Approach
There is a different way to deal with linear modeling. It involves neglecting the “standard”
equation completely and dealing only with the rate of change (the slope) itself.
Since slope is a ratio, it can be used as part of a proportion to answer questions. However,
setting up a correct proportion can be very challenging, as you may recall from previous
experience!
Example: Speedy printing charges $23 for 200 deluxe business cards and $35 for 500 deluxe
business cards. Assuming a proportional cost for the number of cards printed, determine the
cost for printing 700 cards.
Solution: First, determine the rate charged, in dollars per cards (since cost depends on
quantity).
cards300
12$
cards200cards500
23$35$
Now, the hard part – what should the other ratio be?
Here’s what a first response might be: cards700
$
cards300
12$ D
If you solve this by “cross-multiplying” (which is the Fundamental Property of Proportions),
you will quickly find the following:
D
D
D
28
300400,8
30070012
So, 700 cards cost $28. The problem: how can 700 cards cost less than 500??
The issue here is that the rate we used was computed based on the given price data, so our
unknown ratio has to be based on that data as well.
On the 120th day, the
plant reaches 6 feet.
47
That is, we need to measure 700 cards against what we know, and find the cost that adds to
what we know.
Here’s a correct set-up: 700 cards is 500 more than 200 cards, so the ratio cards500
$D will help
us find how much more 700 cards cost than 200 cards. Proceed as before to solve:
D
D
D
D
20
300000,6
30050012
cards500
$
cards300
12$
Practice
1. A retailer of television sets reports that during a 3-month period in which they spent $9,500
on advertising, they sold 16,500 sets. In the next period they spent $22,000 and sold 18,250
sets. What would the estimated sales be if the advertising budget was raised to $100,000?
29,170 sets
2. In 1975, about 450,000 youths participated in amateur softball. By 1995, that number had
jumped to approximately 1,350,000. How many would you estimate are participating in 2010?
2,025,000 in 2010
This tells us that 700 cards cost $20 more than 200 cards; that
is, $43.
You can check that basing the computation on 500 cards
costing $35 will give the same final answer.
48
A polynomial is an expression in which constants and variables are combined using (only)
addition, subtraction, and multiplication. Thus, 547 2 xx is a polynomial; x
2 is not.
Examples: xxxx 61193 234 , 195127218 2345 xxxxx , 4
477
529
23 xxx
Every polynomial with one variable is equivalent to a polynomial with this form:
01
2
2
3
3
2
2
1
1 axaxaxaxaxaxa n
n
n
n
n
n
The constants (numbers) naaaa ,,,, 210 are called the coefficients of the polynomial. The
constant na (from the first term) is called the leading coefficient and 0a is called the constant.
Each summand i
i xa of the polynomial is called a term.
The term of the leading coefficient is called the leading term.
A polynomial with one, two or three terms is called monomial, binomial or trinomial
respectively.
The exponent i is called the degree of the term.
The degree of the polynomial is the degree of the leading term. Polynomials are often named
by degree:
Degree Name Example
0 Constant 103
1 Linear 2511 x
2 Quadratic 201011 2 xx
3 Cubic 12968 23 xxx
4 Quartic 67542 234 xxxx
5 Quintic 127864 2345 xxxxx
One of the remarkable facts observed in a calculus course is that pretty much any (every!)
quantity can be modeled by a polynomial of some degree. Our objective for the remainder of
the semester is to familiarize ourselves with the arithmetic of polynomials.
Addition and Subtraction
We can immediately consider addition and subtraction of polynomials using two familiar
words.
Other names for higher degrees exist.
Look them up if you’re curious!
49
SIMILAR TERMS
That is to say, we’ve already done it, at some level.
For example,
2356346245 614533112104132211938 xxxxxxxxxxxx
can be simplified as:
934310
312212116914104358313
23456
23456
xxxxxx
xxxxxx
The similar powers group together, and coefficients are combined. That’s IT!
Since polynomials may be composed of a large number of terms, it is fairly common to “name” a
polynomial with a single variable label out of convenience.
For example, in the example above,
2356346245 614533112104132211938 xxxxxxxxxxxx
could be named P + Q – R as long as we say that:
P = 2211938 245 xxxx and
Q = 311210413 346 xxxx and
R = 2356 61453 xxxx
Exercises:
Based on the named polynomials above, simplify:
a. Q – P + R b. 2Q – 3R c. 5P + 2R
50
Solutions:
a. 53233247316 23456 xxxxxx b. 6224182281517 23456 xxxxxx
c. 11055572815506 23456 xxxxxx
Exponentiation is a process generalized from repeated multiplication.
Exponentiation involves two numbers, the base and the exponent. The exponent is normally
written as a superscript to the right of the base.
For example, 024,14444445 and 72933333336
It is also true that xxxxxxxxx 8
Notice, though, that xxxxxx 44 5 ; the coefficient 4 is not part of the base and thus not
subject to the power.
However, if it is in the base, then 55024,1444444 xxxxxxx
Exponentiation in this purely algebraic sense satisfies the following laws:
I. 10 x II. xx 1
Anything to the zero power is 1. Anything to the first power is itself.
III. nmnm xxx IV.
nmn
m
xx
x
Product of similar bases: sum of powers. Quotient of similar bases: difference of powers.
V. mnnm xx VI. mmmyxxy
Base to a power, to a power: product of powers. Product of bases to a power: distribute power.
VII. m
mmm
xxxx 11
1
LAWS OF EXPONENTS
Negative power: reciprocate base, opposite power.
51
These laws are only true if both sides are defined. In particular, 0
x is undefined for all
nonzero x, and the expressions 0
0 and 00 are indeterminate, which is definitely undefined.
Simplifying algebraic expressions, particularly polynomial expressions, which involve
multiplication and division is virtually impossible without confident understanding and
application of these laws.
It takes practice to become familiar enough with them to move forward. That is our next goal.
Examples:
1. 232 xyyx
Solution: 74622232 yxyxyxxyyx
2. 3524
4
2
83
cbac
ba
Solution:
3. 36
m
Solution: First: 3
3
6
16
mm
. Then it follows that:
3333 216
1
6
1
6
1
mmm
4.
2
62
35
8
4
vu
ts
Solution: Simplify inside parentheses first:
2
63
252
62
35
28
4
vt
us
vu
ts
Then we find: 126
410
1262
4102
63
25
422 vt
us
vt
us
vt
us
738
812
153812
12
156
8
32123
4
524
2
833524
4
2
83
cbca
cba
a
cb
c
ba
a
cb
c
bacba
c
ba
52
5.
3
2
4
3
6
r
q
Solution: First, simplify inside parentheses:
3
42
3
2
4 2
3
6
qrr
q
Then, reciprocate the exterior negative exponent cleverly:
3423
42 2
2
qr
qr
Then, it’s “quick:” 822
126
3
126342 qrqrqr
6. 30523 3154 PPQP
Solution: First,
323
30523 314
13154 P
PPPQP
This yields:
3
3
6
33
62
3
23 16
2727
16
13
4
131
4
1
PP
PP
PP
P
7.
3
24
2
3
2 12
3
q
p
pqq
p
Solution: Recalling that division is reciprocal multiplication… (…and the previous examples!)
3
64
6
4
3
64
62
43242
3
23
24
2
3
2
129123123
12
3 p
qpq
q
p
p
qpq
q
p
p
qpq
q
p
q
p
pqq
p
This goes on as: 108108129
42
63
105
3
64
6
4 qp
qp
qp
p
qpq
q
p
Exercises: Simplify.
a. 342 25 xyyx b. 28
74
17
51
ba
ba c.
243
2
6
p
nm
Solutions: a. 7310 yx b. 12
93
a
b c.
8
26
9n
pm
Zero power = 1
53
You have begun the study of multiplication and division of polynomials by examining the so-
called laws of exponents. Those are immensely important. Now, the next level.
Example: 63 xx
Solutions:
1. VISUALIZE IT!
2. DISTRIBUTE IT! Two distributions required:
189
1863
6363
633
63
2
2
xx
xxx
xxxx
xxx
xx
Which do you prefer? (Have you heard of FOIL*?) I prefer to “merge” the messages and just
advise:
For example, here’s how I might simplify: 2531342 223 xxxxx :
2531342 223 xxxxx
21512
3123532
3322
452
42
32
423
253
22
33
2 xxxxxxxxxxxxxxxxx
2523621539283204123441056 xxxxxxxxxxx
221032542256 xxxxx
WHEW!
* - What, exactly, is FOIL?
6 x
Area:
Area:
x
3
Area:
6x
Area:
3x
Total area =
+
+
All terms of the first factor distribute to all terms of the second factor.
54
FOIL stands for:
FIRST – OUTER – INNER - LAST
It is meant to describe how to approach multiplication of two binomials, by pairing the terms as
products.
Example: 129 xx
FIRST: The first term of each is an x, and 2xxx
OUTER: The outer terms are x and 12, and that product is 12x.
INNER: The inner terms are –9 and x, and that product is –9x.
LAST: The last terms are –9 and 12, and that product is –108.
Now, combine these components, to complete the word FOIL:
1083108912 22 xxxxx
Therefore, 1083129 2 xxxx
The problem with FOIL is that it doesn’t work for anything other than products of binomials.
Four parts, four letters…what are you going to do with something larger?
That’s why I mention it only as a footnote…because you can do better.
This will definitely take some practice…
55
Exercises:
a. 2244 xxx b. 7392252
xxx
c. 2555 2 xxx d. 96432 2 xxx
Solutions:
a. 4x – 20 b. 67719 2 xx c. 1253 x d. 278 3 x
56
Multiplication of polynomials, continued
(Still the same) Example: 63 xx
Solutions: 3. Formulas: some examples…
222
222
2
2
2
2
Products" Special"
Products General
Squares PerfectA.
BABABA
BABABA
ABxaBAbabxBbxAax
ABxBAxBxAx
3322
3322
22
Cubes of Difference and SumC.
Squares of DifferenceB.
BABABABA
BABABABA
BABABA
Note: Be careful to not make the following special mistakes!
333
333
222
222
AxAx
AxAx
AxAx
AxAx
…et cetera…
Finally… 63 xx
63 xx , referring to the first General formula, ABxBAxBxAx 2 :
Noting A = 3 and B = 6, we have 1896363 22 xxxx
Note: This is an awful lot of fuss compared to the method of full distribution we have
discussed.
In fact, formulas for multiplication of polynomials are generally not very often employed nor
memorized for practical use, at least not in this way…
…but stay tuned for how they are used, very practically.
Example: 7272 xx
This one is arguably the most
significant and widely known.
You should memorize it.
57
Solution: This is a product of conjugate factors, which always gives the difference of
squares:
Noting A = 2x and B = 7, we have 49472 222 xx
Example: 3896 xx
Solution: Referring to the second General formula, ABxaBAbabxBbxAax 2
Noting a = 6, A = – 9, b = – 8 and B = 3:
2790482718724839368986 222 xxxxxx
Example: 2211 xx
Solution: A pair of Perfect Squares:
222
222
2
2
BABABA
BABABA
In our cases, A = B = 1.
xxxxx
xxxxxxxx
41212
1212112112
22
222222
Practicing multiplication of polynomials – whether using FOIL or something “larger” – should be
your goal at this point.
Difference of squares of terms
Conjugate factors
■ All terms of the first factor distribute to all terms of the second factor.
■ Everything distributes to everything
■ ______________________________________________________________________________ (insert your much-more-clever phrase here)
58
A related topic to division, which now becomes our focus, is factoring.
■ To factor an expression means to write it as a product of two or more other expressions.
This means “un-multiplying”; recall that the word “factor” is also used to name parts of
multiplication.
For example, we know that 4624 . Here, 6 and 4 are called factors, and 24 is called the
product. 46 is called a factorization of 24. There are eight integer (positive and negative, not
fractions) factorizations of 24 – can you find the rest?
Just from understanding what factoring means, we get a crucial fact:
You can always check to see if you have factored correctly by multiplying.
The first “method” concept of factoring is anti-distributive in nature…
Whenever two or more terms have a common factor, that factor can be “un-multiplied.”
Example: xxx 5225222104 A common factor of 2 is factored.
■ A working definition of a Greatest Common Factor (GCF) could be:
The lowest power of each variable shared by all terms
and
The greatest number dividing evenly into all coefficients/constants.
Example: 45352635634 70491428 zyxzyxzyzyx
7 is a the largest number that goes into all coefficients evenly.
x: we don’t have all terms involved, so x is not part of this GCF.
y: the lowest power among all terms is 2.
z: the lowest power among all terms is 3.
To complete the factorization, “remove” the GCF and leave behind “the rest” in parentheses:
GCF:
327 zy
(or, 327 zy )
59
zyxzxyyzxzyzyzy
zyx
zy
zyx
zy
zy
zy
zyx 332633432
7
70
7
49
7
14
7
2832 1072477 32
453
32
526
32
35
32
634
or
zyxzxyyzxzyzyzy
zyx
zy
zyx
zy
zy
zy
zyx 332633432
7
70
7
49
7
14
7
2832 1072477 32
453
32
526
32
35
32
634
Notice how I used division in each term to find what was “left inside” the parentheses.
Exercises: Factor these expressions by using a GCF among all terms.
a. 42343 24 bzazba b. 22222 yxxyyx
Solutions: a. zabbza 332 22 b. xyyxxy 2
c. 322343 61824 bababa d. 754254 14228 yxyxyx
Solutions: c. baabba 346 222 d. 33242 71142 yxyxyx
e. 1251 2222 xxxx f. 56642424 xxxx
Solutions: e.
22
222
51
251
xx
xxx
f.
18724
4224
254
254
xxxx
xxxx
g. ddd 396 23
Solution: g. 1123
1323 2
ddd
ddd
■ Grouping: A method of factoring that involves recognizing GCFs among pairs of terms.
Notes:
It’s best for expressions with an even number of terms
Descending (or ascending) order can help here.
60
Examples:
1. 4267 23 xxx First, dissociate (“group”) into pairs:
23 7xx 426 x Now find GCFs:
2x + 6 Factor:
72 xx 76 x Final Factorization:
67 2 xx That is the end of the grouping and both factors are prime.
2. 5511153 23 xxx First, group into pairs:
23 153 xx 5511 x Now find GCFs:
23x + 11 Factor:
53 2 xx 511 x Final Factorization:
1135 2 xx That is the end of the grouping and both factors are prime.
Exercises:
Factor by grouping.
h. xxx 318848 32 i. 70514 23 xxx j. xbaxab 25451018
Solutions:
h. 386 2 xx i. 1452 xx j. xba 5952
3. 84189 23 xxx First, group into pairs:
23 189 xx 84 x Now find GCFs:
29x + 4 Factor:
61
29 2 xx 24 x Factorization:
492 2 xx That is the end of the grouping, not the Final Factorization.
For a Final Factorization, one must rearrange and recognize:
22 942492 xxxx
xxxxxxx 32322322942222
Remember the Difference of Squares. REALLY. It’s everywhere, perhaps way too often.
Difference of Squares: BABABA 22
4. 7289 235 xxx First, group into pairs:
35 9xx 728 2 x Now find GCFs:
3x + 8 Factor:
923 xx 98 2 x Factorization:
89 32 xx That is the end of the grouping, not the Final Factorization.
Again, for a Final Factorization, one must recognize:
422332389 2332232 xxxxxxxxx
The Difference of Cubes (and Sum of Cubes) are less common than the Difference of Squares,
but you’d better recognize them when they appear!
Exercises:
k. xxx 736327 32 l. 101523 23 xxx m. 4014207 23 xxx
Difference of squares!
Difference of cubes! Difference of squares!
Now all factors
are prime.
62
Solutions:
k. xxx 731313 l. 2352 xx m. 20722 xx
The Difference of Squares and Difference of Cubes are product formulas (remember?). Are
product formulas important? Clearly they are to some people, or they wouldn’t have persisted
to this day.
Here’s one that deserves some particular attention:
ABxBAxBxAx 2
This works well for multiplying, as we know, but it might be even better for factoring!
What it says is this:
■ Given a quadratic trinomial of the form DCxx 2 , it factors into NxMx provided
that:
M + N = C and MN = D
Example: Factor 1492 xx
Notice C = 9 and D = 14. Let’s start by factoring D.
14 factors as 141,72,141 and 72 . Of these pairs of numbers, only one adds to 9: 2
and 7.
Therefore I conclude that 721492 xxxx . Check it if you’re not convinced!!!
Exercises:
n. 5432 xx o. 44152 xx p. 2422 xx
Solutions:
n. 69 xx o. 411 xx p. 46 xx
This method of factoring quadratic trinomials is often called “Reverse FOIL”, which does
accurately describe the logic involved.
63
It works extremely well (and often can be executed mentally) for quadratic trinomials of the
specified form: DCxxDCxx 22 1
What if there is a leading coefficient other than 1? Well, for many years, I believed the only way
to accomplish that task was through the grossly inefficient and frustrating “guess-and-check”
approach.
I WAS WRONG. I never learned the way to handle ALL quadratic trinomials…but I know it now!
■ The “AC” Method
1. Given a quadratic trinomial of form CBxAx 2 , compute AC.
2. Find factors of AC that have a sum of B. Note: There may be no such factors.
3. Split Bx into two terms, with coefficients being the numbers found in step 2.
4. Group the resulting polynomial to complete the method.
5. Check your work by multiplying.
Examples:
5. 1872 xx
1. AC = 18181
2. 729;2918
3. 18292 xxx
4.
29
929
xx
xxx
Final Factorization: 29 xx
6. 483 2 xx
1. AC = 1243
2. 826;2612
3. 4263 2 xxx
4.
232
2223
xx
xxx
Final Factorization: 232 xx
This approach covers all 2Ax cases, whether that leading coefficient is 1 or not…COOL!
64
Exercises:
q. 7132 2 xx r. 3196 2 xx s. 9154 2 xx
Solutions:
q. 712 xx r. 316 xx s. 334 xx
Why do we need to study factoring?
One reason is that is is the natural complement to a study of multiplication, and learning both
concepts is meant to help you retain each of them more effectively.
Perhaps the main goal of factoring, though, is to completely split a polynomial into linear
factors; that is, factors of the form BAx with x limited to the first power only.
Why that way? It’s simple to solve for x: ABxBAx 0
65
Here’s where that solving takes its place in higher power equations:
■ Zero Product Property
If 0NM , then M = 0, N = 0 or both are zero. A zero product must have a zero factor.
If we can factor a polynomial set equal to zero, then (at least) any linear factors will show us
solutions (also called roots) of the polynomial equation we start(ed) with.
■ Solving Polynomial Equations
1. Move all terms to one side using addition/subtraction, making the equation equal zero.
2. Factor the polynomial as completely as possible.
3. Use the Zero Product Property to find solutions from any linear factors.
Examples:
1. 030112 xx
Solution:
A quick factorization:
05630112 xxxx
This means x – 6 = 0 or x – 5 = 0,
and that means x = 5 or x = 6
03051152
and
03061162
…it works!
2. 954 xx
Solution: First, simplify: 9204 2 xx
Then, “move –9” by adding 9 to both sides;
creates 0 on the right side
0999204 2 xx
Next, AC = 36 = 2∙18, and 2 + 18 = 20
09212
0129122
091824 2
xx
xxx
xxx
This means that 2x + 1 = 0 or 2x + 9 = 0,
and that means 21x or 2
9x
3. 010155 345 xxx
Solution:
First, factor the GCF, which is 35x : 0235 23 xxx
66
Then factor the interior polynomial:
0125 3 xxx
Now you can separate these factors to state that 05 3 x or x + 2 =0 or x + 1 = 0,
and that means x = 0 or x = –2 or x = –1
Note: You may indeed wonder why 05 3 x implies that x = 0.
This comes from first dividing both sides by 5 to find that 03 x .
Then consider that if 03 xxxx , the Zero Product Property implies that x must be 0.
When solving these kinds of polynomial equations, it is fine to list the solutions as shown in the
examples, but it is perhaps customary (conventional) to use a more formal notation.
For example, instead of listing “x = 0 or x = –2 or x = –1” for the solutions of Example 3, most
mathematicians would probably write this instead:
0,1,2
This pair of curved braces enclosing the solution values is called solution set notation.
It is typical to list the solution values in ascending order (lesser to greater), but not required.
One more bit of vocabulary:
When solving a polynomial equation set equal to zero, the solutions are often referred to as the
roots of the polynomial (and the equation).
This terminology does relate to a mathematical “operation;” another way to deduce the solution
of 03 x is to take the “cube root” of both sides. The cube root of 0 is 0, since 003 .
Roots may be considered as “un-powers” or “anti-powers” and will be considered in further
detail in an Intermediate Algebra course.
67
■ A rational expression has the form Q
P , where both P and Q are polynomials.
You might say that rational expressions are “polynomial fractions.”
First and foremost, note that a rational expression is undefined if Q = 0, so the roots of Q must
be excluded from input.
Example: 45
1092
2
rr
rr
Analysis:
The expression is not defined if 0452 rr , which can be solved by factoring.
14
0104
014
0452
rorr
rorr
rr
rr
Therefore the values r = – 4 and r = –1 must be excluded.
Next, we say a rational expression is in lowest terms when P & Q share no factors (except 1).
Example: Same one: 45
1092
2
rr
rr
Analysis:
It is also not in lowest terms, for factoring the numerator reveals 14
110
45
1092
2
rr
rr
rr
rr
The binomial r + 1 is a common factor and can be “cancelled:” 4
10
14
110
r
r
rr
rr
Therefore 45
1092
2
rr
rr is undefined for 1,4 and has lowest terms
4
10
r
r
Our first arithmetic with rational expressions will be the (relatively) simple and familiar
operations of multiplication and division.
68
■ Key ideas for multiplication and division:
Lowest terms, cross-canceling, reciprocal, factoring
Basically, the idea is to factor numerators and denominators as completely as possible and
cancel complete factors (in parentheses!) when possible.
Then complete multiplication or division, knowing that division = multiplication by the
reciprocal.
As a student once shared with me, division = KSF = Keep / Switch / Flip.
Example: 43
98
16
1610
187
442
2
2
2
2
2
xx
xx
x
xx
xx
xx
Solution: First, there is a heck of a lot of factoring, which leads to:
14
19
44
28
29
22
xx
xx
xx
xx
xx
xx
Now KSF between the first and second terms:
14
19
28
44
29
22
xx
xx
xx
xx
xx
xx
Cancel where available:
14
19
28
44
29
22
xx
xx
xx
xx
xx
xx =
182
142
xxx
xxx
This answer can be expanded in the numerator and denominator if you wish, but it shouldn’t be
considered as a “required” step. Two “acceptable” answers would then be:
16107
825
182
14223
23
xxx
xxx
xxx
xxx
69
Exercises:
1. 54
5322
2
rr
rr 2.
3
56
7
27183 22
x
xx
x
xx 3.
65
82
12
4062
2
2
2
yy
yy
yy
yy
4. 6114
1216342
23
tt
ttt 5.
15112
3145
3012
1252
2
2
2
q 6.
52
5254 2
n
nn
Solutions:
1. 552
rr 2. 833 xx 3.
21
610
yy
yy
4. t2 5. q
q
6
15 6. 552 nn
To discuss addition and subtraction with rational expressions, we turn to a familiar concept…
The Least Common Denominator of any set of fractions is the Least Common Multiple of all
denominators. The LCM of any set of numbers is the smallest number evenly divisible by all
numbers considered.
This carries over to rational expressions quite logically, with factoring a key component.
Example: 32
2
65 22
xx
x
xx
x
Solution: The denominators factor, respectively, as 23 xx and 13 xx
A common factor of (x + 3) means that the LCD is 123 xxx !
Next, I’ll proceed “as usual” – that is, examine what’s missing to form fractions having the LCD,
multiply numerator and denominator to reach the LCD, then simplify the combined numerators.
2
2
13
2
1
1
23
13
2
2332
2
65 22
x
x
xx
x
x
x
xx
x
xx
x
xx
x
xx
x
xx
x
70
Continued:
123
4
123
242
123
221
22
xxx
x
xxx
xxxxx
xxx
xxxx
This expression is in lowest terms, so that’s the final answer. It looks very complex, but that’s
kind of how fractions are…right?
Let’s try to summarize the idea.
■ To add or subtract rational expressions:
1. Factor all denominators to establish a Least Common Denominator.
2. Make all fractions equivalently have that denominator.
3. Add/subtract numerators only.
4. Factor the resulting numerator (if necessary) to verify the answer has Lowest Terms.
Finally, we consider rational equations, that is, equations containing rational expressions.
The method of choice for solving such equations may begin with a familiar idea: clearing
fractions.
That is, eliminate the fractions involved by achieving a common denominator, and then
“clearing” it to leave only the adjusted numerators to solve with.
And how to solve the resulting numerator-based equation? We already know…
■ Zero Product Property
If 0NM , then M = 0, N = 0 or both are zero.
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Here are the steps I recommend.
■ Solving Rational Equations:
1. Find a LCD for all terms of the equation, and convert each rational expression to an
equivalent one with the LCD.
2. Clear fractions to gain a polynomial equation.
3. Simplify if necessary, then move all terms to one side using addition/subtraction, creating
an equation equal to zero.
4. Factor that polynomial as completely as possible.
5. Use the Zero Product Property to find solutions from any linear factors.
Examples:
1. 2
2
86
1
43 222
xx
x
xx
x
xx
x
Solution: The overall LCD is 124 xxx
With the overall LCD, the equation is:
412
42
124
11
214
2
xxx
xx
xxx
xx
xxx
xx
After clearing fractions, one can proceed as follows:
xxxx
xxxxx
xxxxxx
82122
8212
42112
22
222
Continuing, we move all terms to one side: xxxx
xxxx
8282
82122
22
22
– 6x – 1 = 0
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This result doesn’t even require the Zero Product Property. It is linear, so all we need to do is to
isolate the variable x:
– 6x – 1 = 0
+1 +1 – 6x = 1 61x
2. 1
5
8
14
9
5
xxx
Solution: The overall LCD is 189 xxx
With the overall LCD, the equation is:
891
895
198
1914
189
185
xxx
xx
xxx
xx
xxx
xx
Now that common denominator can be discarded, and only numerators retained.
8951914185 xxxxxx
Now proceed as follows:
360551661759
360551261401440355
72591014875
8951914185
22
222
222
xxxx
xxxxxx
xxxxxx
xxxxxx
Moving the three terms from the left to the right side to create 0 yields:
194180140 2 xx
Notice that the (greatest) common factor of 2 can be removed by dividing both sides by it
immediately:
194180140 2 xx 2
194
2
180
2
14
2
0 2
xx
979070 2 xx
Then, the AC method: AC = – 409 = –7 · 97, and –7 + 97 = 90:
97710
197170
9797770 2
xx
xxx
xxx
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This means that x – 1 = 0 or 7x + 97 = 0, and that means 1x or 797x
Exercises: Solve.
1. xxxx
x
2
81
2
22
2.
6
23
6
122
a
a
aa
Solutions:
1. 3 2. 23