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Chapter 1: Equations and inequalities1.1 apply properties of real numbers
Real Number System
NATURAL
WHOLE
INTEGERS
IRRATIONALSRATIONALS
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Natural Numbers
Counting Numbers1, 2, 3, 4, 5, ...
Example of where you have seen Natural Numbers used.
∞
page 1
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Whole Numbers
0 + All of the Natural Number 0, 1, 2, 3, 4, 5, ...
Example of where you have seen Whole Numbers used.
∞
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Integers
All of the Whole Numbers + all of the opposites of the Natural Numbers
. . . , -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, . . .
Example of where you have seen Integers used.
‐∞ ∞
page 1
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Rationals
• All Natural, Whole, and Integers• Any number you can write as a fraction
where a & b are integers with b≠0
ab
• Any terminating decimal (0.5, 7.13, -6.876, -24.45)
• Any repeating decimal (‐10., 2., 3.6262…, ‐2.12333…)
• Square roots of Perfect Square Numbers (1, 4, 9, 16, 25, ...)
Example of where you have seen Rational Numbers used.
page 1
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Irrationals
• Any number that is NOT Rational• Decimals that do not terminate AND do not repeat• "CRAZY NUMBERS"• Square roots of non-perfect square numbers (√2,
√23, √30, √55)• Can be positive or negative
Example of where you have seen Irrational Numbers used.
page 1
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Graph real numbers on a number line
1 02345 1 2 3 4 5
Graph the real number on a number line
Which list shows the numbers in increasing order?
a) 0.5, 1.5, 2, 0.75, √7 b.) 0.5, 2, 0.75, 1.5, √7
c) 2, 0.75, 0.5, 1.5, √7 d.) √7, 1.5, 0.5, 0.75, 2
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Properties of addition and multiplication
let a, b, and c be real numbers
Property
Closure
Commutative
Associative
Identity
Inverse
Addition Multiplication
a + b is a real number.
a + b = b + a
(a + b) + c = a + (b + c)
a + 0 = a, 0 + a = a
a + (a) = 0
ab is a real number.
ab = ba
(ab)c = a(b c)
1 a = a, a 1 = a
a = 1, a ≠ 0
The following property involves both addition and multiplication
Distributive: a(b+c) = ab + ac
Defining Subtraction and Division
Subtraction is defined as adding the opposite. The opposite, or additive inverse, or any number b is b. If b is positive, then b is negative. If b is negative, then b is positive.
Division is defined as multiplying by the reciprocal. The reciprocal, or multiplicative inverse, of any nonzero number b is 1/b.
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Identify the property that the statement illustrates
Commutative
Associative
Identity
Inverse
Distributive
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Unit analysis and conversions
You work 4 hours and earn $36You travel 2.5 hours at 50 mph. How far did you go?
You drive 45 miles per hour. What is your speed in feet per second?
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1.2 Evaluate and Simplify Algebraic expressions
A numerical expression consists of numbers, operations and grouping symbols. An expression formed by repeated multiplication of the same factor is a power.
73base
exponentPower
Terms and Coefficients
3x2 + 5x + 7
coefficients
variable termsconstant terms
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Evaluating different expressions
1.
2.
3.
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1.3 Solve Linear equations
An equation is a statement that two expressions are equal. A linear equation in one variable is an equation that can be written in the form ax+b = 0 where a and b are constants and a≠0.
example 1:
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example 2:
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example 3:
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Car Wash: It takes you 8 minutes to wash a car and it takes a friend 6 minutes to wash a car. How long does it take the two of you to wash 7 cars if you work together?
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1.4 Rewrite formulas and equationsA formula is an equation that relates two or more quantities, usually represented by variables. Some common formulas are shown below.
Distance
Temperature
Area of a Triangle
Area of a Rectangle
Perimeter of a Rectangle
Area of a trapezoid
Area of a circle
Circumference of a circle
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example 1: The formula for the distance d between opposite vertices of a regular hexagon is below. Where a is the distance between opposite sides. Solve the formula for a.
example 1.5: Write the formula giving the are of a circle in terms of its circumference.
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example 2: Rewrite linear/nonlinear equations
Linear: Solve for y
Nonlinear: Solve for y
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example 3: Solve for y, leave your answer in simplest form
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1.6 Solve Linear Inequalities
Graphing inequalities:
or
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examples: Solving and graphing inequalities
or
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example: A monitor lizard has a temperature that ranges from 18oC to 34oC. Write the range of temperatures as a compound inequality. Then write an inequality giving the temperature range in degrees Fahrenheit.
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1.7 Solve Linear Inequalities
Interpreting Absolute Value Equations
Equation:
Meaning:
Solution:
The distance between x and b is k
or
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Example: Solve absolute value equations
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Example: Check for extraneous solutionsExtraneous Solutions: It is possible for a solution to be extraneous. An extraneous solution is an apparent solution that must be rejected because it does not satisfy the original equation.
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Example: Solve and graph an absolute value inequality
<
>