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MAT028A
TOPICS: ALGEBRAIC EXPRESSIONS
SOLVING EQUATIONS AND
INEQUALITIES
Algebraic Expressions
1.1
Introduction to Algebra
I. You must memorize some vocabulary!
A. “Evaluate” means plug and crunch.
“Algebraic” means you’ll see letters.
“Expressions” means you won’t see “=”.
“Variables” are typically letters that stand for a class of numbers.
II. Translating to Algebraic Expressions
A.
Addition
Sum
Add
Total
Plus
More than
Increased by
B. *
1.2
Subtraction
Difference
Subtract
Minus
Less than*
Decreased by
Take away
Watch Out!
Multiplication Division
Product
Quotient
Multiply
Divide
Times
Of*
Ex: “7 less than 12” Translates to 12-7
Ex: “Of” can imply others things… sum of two numbers
The Real Numbers
I. Keep in mind that each set below includes the sets to its left!
“Natural”
“Whole”
“Integer”
{1,2,3,…} Throw in {0,1,2,…}
Throw in
{…-2,
Throw in all
zero
negative
-1,0,1,2,…} #’s that can
cousins
be written
a/b, b≠0
“Rational”
{ Ex: , 0.2,
0.3,
10%}
Throw in
irrationals
Ex:
Sometimes textbook examples of integers in the real world are a little confusing.
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“Real”
On line
II. Order
A.
Smaller
Left
Larger
Right
<
B. When comparing decimals go place value by place value, left to right.
Ex: 1.273 < 1.28
C. Fraction Trick
? Cross multiply and the fraction closer to the larger # is larger!
III. “Absolute value” is simply distance from zero on the # line so it must be positive.
1.3 Addition of Real Numbers
I. Adding using a number line: All positive numbers go right, all negatives go left.
II. Adding using memorized rules:
Same Sign? Use sign and combine.
Diff. Sign? Use sign of # with larger absolute value and subtract!
II. “Opposites” or “Additive Inverses”
A. Vocab: “Additive inverse” is the more sophisticated expression.
B. On the # line, additive inverses reflect through zero.
-2 0 2
These are additive inverses. They add up to zero.
C. The inverse of the inverse is the # itself! (“I can’t get no satisfaction!”)
Ex. Consider 6. Its Inverse is -6. And its inverse is –(-6) which is 6.
Coming Soon!
An Even # of negative signs gives you a positive #.
An Odd # of negative signs gives you a negative #.
D. The words “Inverse of” can be replaced with this symbol “-“
1.4
Subtraction of Real Numbers
Step 1:
Step 2:
1.5
Get rid of subtraction in one of two ways:
Ex: 3-(-2) = 3+2
Ex: 5-7 = 5 + (-7)
(Subtraction is adding the opposite in both cases.)
Use Addition Rules Above
Multiplication of Real Numbers
I. The Rules: That last rule is hard to visualize, so memorize!
This is Shorthand for
+ + + (POS) (POS) =(POS)
- - +
(POS) (NEG) =(NEG)
Ask your teacher about good guys and bad
(NEG) (POS) =(NEG)
guys if you’re having trouble.
(NEG) (NEG) =(POS)
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II. Suppose you have lots of numbers.
EX: (-2)(-5)(7)(3)(-1)
The product of an even # of negative #’s is positive.
“ “
odd
““
negative.
III. That Sneaky Aunt Sally
Ex: Evaluate
and –x2 when x=2
2
Plug (-1 ∙ 2)
Plug -1 ∙ 22
In Parens (-2)2
-1 ∙ 4 Exp
Exponent 4
-4 Mult
Exponents only hit things immediately next door unless see ( ).
1.6
Division of Real Numbers
I. The Sign Rules: Same as multiplication because division can be seen as a form of multiplication.
Ex: 5 ÷ 6 = 5 ∙
II. “Reciprocals” or “Multiplicative Inverse”
A. Vocab: “Multiplicative inverse” is the more sophisticated expression.
Ex: and are multiplicative inverses. Their product is 1.
B. The inverse of the inverse is the # itself!
III. For a review of fractions and decimals, see 018A and 018B.
1.7
Properties of Real Numbers
I. To simplify expressions or solve equations we create “equivalent expressions” as long as we
follow the rules or “properties.”
II. The Properties and the things they allow us to do.
Name
Addition
Multiplication
Commutative
{Order doesn’t matter so move them for your convenience.}
Associative
{Group as you please.}
Identity
{Identities don’t change things.}
Inverse
{Inverses take you to Identities.}
Distributive
{Distributing combines multiplication with addition.}
A. The multiplicative identity, 1, allows us to build up and reduce fractions.
Ex:
)
Ex:
)
B. Distribution, run backward, allows us to “factor.”
Ex:
Largest #/var that goes into each term. Divide by this to get stuff in parens.
III. “Terms” are the train cars in the train.
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A. The polynomial 5x - 7y + 5 has the terms 5x, -7y, and 5.
B. “Collecting like terms” means combining numbers in front of identical var powers.
These numbers are the “coefficients” and they may be an invisible one!
1.8 Simplifying expressions: Order of Operations
I. “Simplified expressions” have the fewest possible symbols and the lowest possible numbers.
There are many different contexts for simplifying! Use the above statement as a guide.
II. Some books mix up arithmetic with algebra.
A. If you see no letters, it’s arithmetic and Aunt Sally rules.
1. PEMDAS
1st: inside parens, 2nd: exponents, 3rd: × ÷ left to right, 4th: + - left to right.
2. For error control, try this style:
Ex:
aside
B. If you do see letters, it’s algebra. Get rid of the parens and combine like terms.
1. Work from inside out { [ ( ) ] }
2. Once you’ve distributed, you may remove parens.
Solving Equations and Inequalities
2.1
Solving Equations: the Addition Principle
I. Vocab
A. “Equations” are expression that have “=” in there.
B. “Solutions”, when plugged into an equation, make it “true.”
C. “True” means both sides simplify to the same thing.
II. The “Addition Principle” allows you to add the same # (positive or negative) to both sides to get
an equivalent equation.
How to do it:
1. Locate your “=” and your unknown.
2. Since you want your unknown alone remove its neighbor by adding its additive inverse
(or opposite) to both sides. (Write it directly below each number.)
3. Check your result by “plug and chug.”
Ex: t – 3 = 19
+ 3 +3
t
= 22
Check: 22 - 3=19?
19 = 19
2.2
Solving Equations: the Multiplication Principle
I. The “Multiplication Principle” allows you to multiply both sides by the same # to get an
equivalent equation.
How to do it:
1. Locate your “=” and your unknown.
2. Since you want your unknown alone, remove its neighbor by multiplying both sides by
its multiplicative inverse (or reciprocal). (Write it next to each side.)
3. Check your result with “plug and chug.”
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Ex
(
) (- ) x = (
) (-
)
2
x=Check:
?
=
2.3
Using the Principles Together
I. The properties (commutative, associative, identity, inverse, distribution), the principles (+,×),
and Aunt Sally all combine to allow you to solve equations.
Dogs
Cats
Mice
Insects
D istribute and, throughout, collect (newly released) like terms.
C* lear fractions and decimals.
M ove terms so unknowns all to one side.
I solate using reciprocal of # next to unknown.
C heck.
II. Clearing Fractions Trick*
Step 1: Identify LCD but don’t use it in normal way.
Step 2: Write ( ) or (LCD) next to each term.
Step 3: Cross cancel.
Step 4: Clean up and finish.
Ex:
(Note: Since x-term is unique, move its neighbor.)
-2
-2
III. “Weird” solutions
A. No solution:
Ex:
(This makes no sense!)
B. Infinitely-many solutions:
Ex:
(When sides are same, any value for x will solve it!)
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2.4
Formulas
Same Dogs, Cats, Mice, and Insects as above! (Distribute/Clear/Move/Isolate)
2.5
Applications with Percent
I. The Portion Triangle
25% (A Percent is usually multiplied by something.)
0.25
II. To solve most percent problems, you need one of these techniques.
A. Translation
“is”… =
“of”…
“what”…
“percent”… 0.01
Some books are sloppy about translating “percent.”
B.
The is a “proportion” and it can be “cross multiplied.”
III. Three Possible forms here:
A. What is 14% of 250?
x =(14)(0.01)× 250
B. 30 is what percent of 640?
30= x (0.01)× 640
C. 32 is 9 percent of what?
32=(9) (0.01)× x
2.6
Applications and Problem Solving
As always, Please Let Frankie Sing! (pictures, labels, formulas, substitutions)
I. Tricky “after” percent problems- The stuff above won’t help you with this!
Ex: After a 7% weight gain, cat weighs 26lbs. How much before?
Set up:
{If weight loss, use subtraction!}
II. “Consecutive integers” must be named
Consecutive odd or even integers must be named
III. Info given on “whole” and “pieces”- draw pictures!
Ex: 3-foot board cut into two pieces; one piece 1 ft longer than other…
x+1
x
3
IV. The measure of three internal angles on a triangle add up to 180 degrees.
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V. Etc: Geometric formulas are often inside cover of texts.
Average =
Expenses are often= Fixed Expense + (Variable Expense) (#units)
2.7
Solving Inequalities
I. You know that -2 < 3. If we multiply or divide both sides by -1 we have 2 -3 but the
inequality must flip.
II. Solve these just as you would an equation except…
A. Flip inequality if multiply or divide by a negative number.
B. Graph of solution looks different: Typically lines not dots
Ex: x>-2
x≤4
Look at
<
>
<
>
circles!
-3-2-1 0 1 2 3 4 5
1 23 4 5
C. Inequalities must be flipped carefully: x < 3
Pivot through inequality to get 3 > x
2.8
Applications and Problem Solving with Inequalities
I. Vocab from text:
Bill is at least 21
b 21
At most 5 students
n≤ 5
Cannot exceed $21000 r 21,000
Must exceed 15 mph
s>15
Less than 50 lbs
w<50
More than 200 miles
d>200
Between 90 and 100
90<t<100
No more than 90lbs
w 90
No less than 8.3
s 8.3
II. When doing word problems you may wish to use “=” and then insert your inequality at the end.
As always with word problems, PLEASE LET FRANKIE SING! (PICTURE, LABEL, FORMULAS, SUB)
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