Whole Numbers Are the whole numbers with the property of addition a group?
EQUATIONS AND INEQUALITIES A2H CH 1 APPENDIX. Whole: 0, 1, 2, 3….. Integers: Whole numbers...
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Transcript of EQUATIONS AND INEQUALITIES A2H CH 1 APPENDIX. Whole: 0, 1, 2, 3….. Integers: Whole numbers...
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EQUATIONS AND INEQUALITIES
A2H CH 1
REAL NUMBERS AND NUMBER OPERATIONS
ALGEBRAIC EXPRESSIONS AND MODELS
SOLVING LINEAR EQUATIONS
REWRITING EQUATIONS AND FORMULAS
PROBLEM SOLVING
LINEAR INEQUALITIES
ABSOLUTE VALUE EQUATIONS AND INEQUALITIES
APPENDIX
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Subsets of Real Numbers
Whole: 0, 1, 2, 3…..
Integers: Whole numbers negative and positive. … -2, -1, 0, 1, 2,
3…
Rational Numbers: Can be written as a fraction. Either terminates or repeats.
Irrational Numbers: Can not be written as a fraction. Goes on forever without
repeating.
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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Classify each of the following. e. -5 f.
g. -4.8 h.11
INTEGER RATIONAL
RATIONAL IRRATIONAL
43
7
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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• Graph the real numbers , -1.8, , and -.25 and then order them from least to greatest.
5
2 3
-2 3-1-3 0 1 2
-.25-1.85
23
-1.8, -.25, , and .35
2
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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• Use the symbol < or > to show the relationship.
b. -5 and -7
a. -4 and 1
-4 < 1 or 1 > -4
-7 < -5 or -5 > -7
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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Example 3:
• Here are the record low temperatures for five Northeastern states.
Write the temperatures in increasing order.Connecticut -32F
Maine -48F
Maryland -40F Which states have record lows below
-40F?New Jersey -34F
Vermont -50F
Maine and Vermont
Vermont, Maine, Maryland, New Jersey, and Connecticut
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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PROPERTIES OF ADDTION AND MULTIPLICATIONLet a, b, and c be real numbers.
Property AdditionMultiplication CLOSURE a + b is a real number ab is a real number COMMUTATIVE a + b = b + a ab = ba ASSOCIATIVE (a + b) + c = a + (b + c) (ab)c = a(bc) IDENTITY a + 0 = a, 0 + a = a a1 = a, 1a = a INVERSE a + (-a) = 0 a (1/a) = 1 (a ≠0)
The following property involves both addition and multiplication. DISTRIBUTIVE a(b+ c) = ab + ac
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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616
235235
14
14
6336
055
.
)()(.
.
.
.
E
D
C
B
A INVERSE PROPERTY of ADDITION
COMMUTATIVE PROPERTY of ADDITION
INVERSE PROPERTY of MULTIPLICATION
ASSOCIATIVE PROPERTY of ADDITION
IDENTITY PROPERTY of MULTIPLICATION
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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State the property.
a. 8 + -8 = 0 b. 91 = 9
c. ab = ba d. 8 + (2 + 6) = (8 + 2) + 6
INVERSEaddition
ASSOCIATIVEaddition
COMMUTATIVEmultiplication
IDENTITYmultiplication
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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DefinitionsThe opposite (or additive inverse) of any number a is –
a.
The reciprocal (or multiplicative inverse) of any nonzero number a is 1/a.
Subtraction is defined as adding the opposite.
Division is defined as multiplying by the reciprocal.
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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VOCABULARY
ADDITION
+SUBTRACTION
-MULTIPLICATION
XDIVISION
÷
SUM DIFFERENCE PRODUCT QUOTIENT
Decreased by
Divided by Of
Less Than
More Than Minus
Times
Increased by
For each of these operations, use the numbers in the same order they appear in
the problem
The only exception is LESS THAN. When writing a mathematical expression using LESS THAN, use the numbers in reverse
order.
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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a. The difference of -3 and -15 is:
b. The quotient of -18 and 2 is -9:
c. Eight less than a number is twelve.
-3 + 15 = 12
= -9
128 x
2
18
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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Definitions
Writing the units of each variable in a real-life problem is called unit analysis. It helps you to
determine the units for the answer.
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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Give the answer with the appropriate unit of measure.
a. 685 feet + 225 feet
b.
c.
d.
602.25
1km
hourshour
910 feet
2.25 dollars per pound
135 km
94
dollarspounds
66 60sec 60min 11sec 1min 1 5280
feet milehour feet
45 miles per hour
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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• You are exchanging $500 for French francs. The exchange rate is 6 francs per dollar. Assume that you use other money to pay the exchange fee.
a. How much will you receive in francs?
b. When you return you have 270 francs left. How much can
you get in dollars? Assume that you use other money to pay the exchange fee.
6500 3000
1francs
dollars francsdollar
1270 45
6dollar
francs dollarsfrancs
Real Numbers & Number Operations
A2H CH 1 Equations and Inequalities MENU
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Exponents can be used to represent repeated multiplication.
EXPONENT
25 = 2 2 2 2 2 =
BASE FACTORS
An exponent tells you how many times the base is used as a factor.
Algebraic Expressions & Models
A2H CH 1 Equations and Inequalities MENU
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• Evaluate the power.
b. -34
a. (-3)4
(-3)(-3)(-3)(-3) = 81
-3333 = -81
Algebraic Expressions & Models
A2H CH 1 Equations and Inequalities MENU
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An order of operations helps avoid confusion when evaluating
expressions.P
E
MD
AS
P Parenthesis. Compute everything inside the parenthesis.
E Exponents. Evaluate powers.
MD Multiplication/Division. Multiply and divide from left to right
AS Addition/Subtraction. Add and subtract from left to right.
Algebraic Expressions & Models
A2H CH 1 Equations and Inequalities MENU
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• Evaluate using order of operations.
a. b. c.38 5(1 ( 3)) 4 6
2 ( 3)
2 35 6(2 ( 1))
d. e.22 4 8 3
57 2
2 10( 10) 3 ( 10)
2
Algebraic Expressions & Models
A2H CH 1 Equations and Inequalities MENU
312 19 2
125-7.5
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• Evaluate using order of operations.
e. Evaluate 2x3 + 3x2 – x + 27 when x = -4
2(-4)3 + 3(-4)2 – -4 + 27 = -49
-4(-3)2 + 6(-3) – 5 = -59
d. Evaluate -4x2 + 6x – 5 when x = -3
Algebraic Expressions & Models
A2H CH 1 Equations and Inequalities MENU
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Algebraic Expressions & Models
A2H CH 1 Equations and Inequalities MENU
Consider the expression 5x3 – 2x + 9.
The parts that are added or subtracted together (5x3, -2x, and 9) are called terms.
The numbers if front of the variables (5 and -2) are called coefficients of the variables.
When a term is only a number, it is called a constant (9). Terms such as 5x3 and -7x3 are like terms because they have the same variable part. Constant terms such as -6 and 4 are also like terms. You can only combine (add or subtract) like terms.
To combine like terms, add or subtract the coefficients and leave the variable and its exponent the same.
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• Simplify the expression. a. -10(8 – y) – (4 – 15y) b. 4 – 3(x – 9) – (x + 1)
c. 3x + 10 – 12x – (-4) d. 3(x – 2) – 5x(x – 8)
Algebraic Expressions & Models
A2H CH 1 Equations and Inequalities MENU
25 - 84y -4x + 30
-9x + 145x2 + 43x - 6
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Evaluate the following expression for the given values:
24 23 2 , 5, 3
xxy y x y
y
24 23 2 , 5, 3
xxy y x y
y
5 5 ( 3)
( 3)
2245 18
3
( 3)
2245 18
3
1673
55.6
Algebraic Expressions & Models
A2H CH 1 Equations and Inequalities MENU
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Solving Linear Equations
4672377 xx )(
17
4621677 xx
171277 x17
x1260 12 12
x5
A2H CH 1 Equations and Inequalities MENU
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305211 xxx5
16
x5
30216 x22
3216 x16
2x
Solving Linear Equations
A2H CH 1 Equations and Inequalities MENU
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30
2
3
3
74
2
1
5
3xx
Solving Linear Equations
A2H CH 1 Equations and Inequalities MENU
2
7
3
74
10
3
5
3 xx
10570120918 xx
1057012918 xx
x52234
x52
2342
9
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Solving Linear Equations
A2H CH 1 Equations and Inequalities MENU
82 x93 x
x2
If the perimeter is 153, find the length of each side
15328293 xxx15317 x1547 x22x
44
57 52
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Rewriting Equations & Formulas
A2H CH 1 Equations and Inequalities MENU
Solving Literal Equations:
hftz
2432 yx
A Literal Equation is an equation with more than 1 variable
We frequently manipulate literal equations when working with formulas, or changing an equations “form”.
ExampleshbA 3
3
4rV
321
r
MMgF
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Rewriting Equations & Formulas
A2H CH 1 Equations and Inequalities MENU
Find the value for y in this equation when x is 3.
hftz
243
2 y
x3
243
6 y
183
y
54y
Now, complete this table:
X Y
9
12
-4
61
-3.8
When you are asked to do something like this, it is usually easier to change
the equation.
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Rewriting Equations & Formulas
A2H CH 1 Equations and Inequalities MENU
This equation is written in standard form. Rewrite it in slope-intercept form. (solve for y)
hftz
243
2 y
xNow, complete this table:
X Y
9
12
-4
61
-3.8
x2 x2
xy
2243
3 3
xy 672
18
0
96
-294
94.8
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Rewriting Equations & Formulas
A2H CH 1 Equations and Inequalities MENU
Solve the equation for the indicated variable.
hftz
whwlV ; rlrBA ;
hwl
V
whl
V
lrBA
rl
BA
ll
h h
B B
ll
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a. You have $55 to buy digital video discs (DVDs) that cost $12 each. Write an expression for how much money you have left after buying n discs. Evaluate the expression when n = 3 and n = 4.
VERBAL MODEL
LABELS
ALGEBRAICMODEL
–
Price per DVD
Amount to Spend
Number of DVDs
bought
–
12 dollars per DVD
55 dollars
n DVDs
55 – 12n
Problem Solving
A2H CH 1 Equations and Inequalities MENU
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• a. Write an expression for the total monthly cost of phone service if you pay a $5 fee and 8¢ per minute. Find the cost if you talk 6 hours during the month.
VERBAL MODEL
LABELS
ALGEBRAICMODEL
+
Price per minute
Initial Fee
Number of Minutes
+
.08 dollars per minute
5 dollars n Minutes
5 + .08n
Problem Solving
A2H CH 1 Equations and Inequalities MENU
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Linear Inequalities
A2H CH 1 Equations and Inequalities MENUAPPENDIX
ABRIDGED ALGEBRA I
INEQUALITY NOTES:
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Linear Inequalities
A2H CH 1 Equations and Inequalities MENUAPPENDIX
Tougher inequality problems:
8 53x
Solve and graph:
0 9
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Linear Inequalities
A2H CH 1 Equations and Inequalities MENUAPPENDIX
Tougher inequality problems:
6 4 14 2 14x and x Solve and graph:
0 3 7
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Linear Inequalities
A2H CH 1 Equations and Inequalities MENUAPPENDIX
Tougher inequality problems:
4 10 2 2x or x Solve and graph:
06 1
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Linear Inequalities
A2H CH 1 Equations and Inequalities MENUAPPENDIX
Tougher inequality problems:
1 5 2 3x Solve and graph:
0 2 4
3-part inequalities must be “AND”
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Linear Inequalities
A2H CH 1 Equations and Inequalities MENUAPPENDIX
Tougher inequality problems:
1 9x
What’s wrong with this:
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Linear Inequalities
A2H CH 1 Equations and Inequalities MENUAPPENDIX
Tougher inequality problems:
4 7x and x
Solve and graph:
No solution
0
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Linear Inequalities
A2H CH 1 Equations and Inequalities MENUAPPENDIX
Tougher inequality problems:
4 7x or x
Solve and graph:
ALL REAL NUMBERS
0 4 7
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Absolute Value Equations & Inequalities
A2H CH 1 Equations and Inequalities MENU
Absolute Value is how far a number is from zero on a number line.
5 This means: How far f rom zero is 5?
0 5-5
How far f rom zero is -5?
5
5 5
main
ABRIDGED ALGEBRA I
Absolute Value NOTES:
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Absolute Value Equations & Inequalities
A2H CH 1 Equations and Inequalities MENU
Solve this absolute value inequality:
3 7 1 5x
NO SOLUTION
3 7 4x
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Absolute Value Equations & Inequalities
A2H CH 1 Equations and Inequalities MENU
Solve this absolute value inequality:
3 7 1 5x
ALL REAL NUMBERS
3 7 4x
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APPENDIX
A2H CH 1 Equations and Inequalities MENU
OPENERSASSIGNMENTSEXTRA PROBLEMS
APPENDIX
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REVIEW
A2H CH 1 Equations and Inequalities MENUAPPENDIX
Properties of equalityEvaluateAbsolute Value InequalityLiteral EquationsOrdering NumbersSimplifySolve (with fractions)SolveAbsolute Value Inequality
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Properties of addition and multiplicationIdentify the property illustrated: 1. 4 ( 4) 0
2. ( ) ( )
3. 5 0 5
4. 5 1 5
15. 5 1
56. 3(2 1) 6 3
7.
x y z x y z
x x
x y is a real number
Inverse Property (addition)
Associative property (addition)Identity property
(addition)
Identity property (multiplication)
Inverse property (multiplication)
Distributive Property
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A2H CH 1 Equations and Inequalities MENUAPPENDIX
Closure property (addition)
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Evaluate the following for the values
x=4 and y=9
227. 3 ( )
1x
x yy
44 9
9
224
3 (4 9)9 1
168
5
3 2 25
6 25
31
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SOLVE FOR X
8. 9 2 3x
9 2 3 9 2 3x x 2 6x
3x
2 12x
6x
3 6
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99 99
22 22
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Write this equation in function form (solve for y)9. 4 2 10x y
2
10.By
p gmA
4 4x x
2 10 4y x 2 2
10 42
xy
5 2y x
p p
2Bygm p
A
A AB B
2 Ay gm p
B
Ay gm p
B
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11. Write the following numbers in increasing order:
8137, , , 5, , 17 17
3.14159
0.42852.2360
4.7647
37 1 5 81
177
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Simplify the following.
3 312. 3( 2 5) 2( )x x x 3 36 15 2 2x x x 38x 2x 15
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12
12
SOLVE FOR X.
2 1 5 313. 5
3 2 6 4x x
2 10 5 36 3 6 4
x x
1 10 5 33 3 6 4
x x
4 40 10 9x x 40 6 9x 49 6x
496
x
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SOLVE FOR X.
14. 3(2 7) 5( 3 1) 12x x
6 21x 15 5 12x
6 21 15 7x x
21 9 7x
28 9x
289
x
6x 6x
77
9 9
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Write an absolute value inequality to describe the following scenario:
15. On the Eisenhower expressway (I-290), you must
drive at least 45 mph but less than 65 mph. 55 10x
The average of the extremes The tolerance (difference between the average and extremes)
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One of these inequalities is NO SOLUTION, the other is ALL REAL NUMBERS, can you
identify which one is which?3 8 9x
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A2H CH 1 Equations and Inequalities MENUAPPENDIX
3 8 9x
ALL REAL NUMBERS
NO SOLUTION