chapter 1 equations and inequalities. 2 minute vocabulary activity take 2 minutes to define and/or...
TRANSCRIPT
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Chapter 1Equations and Inequalities
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2 Minute Vocabulary Activity•Take 2 minutes to define and/or give an
example of each vocabulary word below…VariableVariable CoefficientCoefficient
Monomial Monomial ConstantConstantDegreeDegree Order of operationsOrder of operations
TermTerm Like termsLike termsBinomialBinomial PolynomialPolynomial
TrinomialTrinomial
1.1
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How do you use the order of operations to correctly evaluate expressions?•Example 1:
▫Evaluate (x – y)3 + 3 if x = 1 and y = 4
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•Example 2:▫Evaluate 8xy + z3 if x = 5, y = -2, and z =
-1 y2 + 5
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•Example 3:▫Find the area of a trapezoid with base
lengths of 13 meters and 25 meters and a height of 8 meters. A= ½ h (b1 + b2)
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What are the different types of numbers?•Real Numbers
▫The numbers used in everyday life, each corresponding to exactly one point on the number line.
•Rational Numbers▫A real number that can be expressed as a
fraction (ratio). The decimal form is either terminating or repeating.
• Irrational Numbers▫Any real number that is not rational.
•Not Real Numbers▫The square root of a negative number
1.2
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Definitions•Natural Numbers (N): counting numbers 1,
2, 3…•Whole Numbers(W): natural numbers plus 0• Integers(Z): whole numbers plus the opposite
of any natural number• Irrational Numbers(I): any number with or
√ where the number under the √ is not a perfect square
•Not real Numbers: any √ where the number under the √ is negative
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Practice
Example 1:Name the sets to which each number
belongsa. √6b. 5c. -2
3
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The Reminders from Algebra I• Properties that you MUST know…
• Commutative: (order changes)▫ (+) a + b = b + a (●) a•b = b•a
• Associative: (groups change but order doesn’t)▫ (+) (a + b) + c=a + (b + c) (●) (a • b)•c= a • (b •c)
• Identity: (after adding or multiplying # is same)▫ (+) a + 0 = a (●) a • 1 = a
• Inverse: (add or multiply the # to cancel)▫ (+) a + (-a) = 0 (●) a • 1/a = 1
• Distributive: (multiply # outside by all inside)▫ a(b+c) = ab +ac
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More Practice
•Example 2Name the property
a. ( -8 + 8) + 15 = 0 + 15
b. ( 5 + 7) + 8 = 8 + (5 + 7)
c. ¼ (4x) = x
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Verbal Expressions to Algebraic Expressions•1. Write an algebraic expression to
represent each verbal expressiona. three times the square of a
number
b. twice the sum of a number and 3
c. the cube of a number increased by 4 times the same number
1.3
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Algebraic to Verbal Sentence
•2. Write a verbal sentence to represent each equation.
a. n + (-8) = -9
b. g – 5 = -2
c. 2c = c2 - 4
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Solving Equations Practice #3
a. a + 4.39 = 76
b. -3d = 18 5
c. 2(2x + 3) – 3(4x – 5) = 22
d. -10x + 3(4x – 2) = 6
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Apply the properties of Equality
•If what is the value of 3n - 3
•If what is the value of 5y - 6
5
983 n
3
825 y
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Solve for a Variable
•The formula for the surface area S of a cone is where l is the slant height of the cone and r is the radius of the bas. Solve the formula for l.
2rrlS
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Write an Equation
•Josh spent $425 of his $1685 budget for home improvements. He would like to replace six interior doors next. What can he afford to spend on each door?
16
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Absolute Value
•For any real number a, if a is positive or zero, the absolute value of a is a. If a is negative, the absolute value of a is the opposite of a.
|a|= a if a >0 |a|= -a if a < 0
1.4
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Work in pairs (speed-date activity)
Evaluate an Expression with Absolute Value
a. 1.4 + |5y – 7| if y = -3
b. |4x + 3| - 3 ½ if x = -2
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Solve an Absolute Value Equationc. |x – 18| = 5
d. 9 = |x + 12|
e. 8 = |y + 5|
f. |5x – 6 | + 9 = 0
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Solve an Absolute Value Equationg. |x + 6| = 3x – 2
h. 2|x + 1| - x = 3x – 4
i. -2|3a – 2| = 6
j. 3|2x + 2| - 2x = x + 3
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Remember those Algebra 1 Properties?•When solving inequalities the properties
all work the same as with equations except…
When you multiply or divide by a negative number you must flip the inequality symbols
Ex: -12x > 96
-12x > 96 -12 -12
x < -8
1.5
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Set-Builder Notation- How to write you answers
• The solution set of an inequality▫ Example
-0.25y > 2-0.25y > 2-0.25 -0.25
y < -8
{y | y < -8}*read the set of all y such that y is less than
or equal to negative 8
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•Example 19
4m
m
-1 0
49 mm
410 m
10
4m
5
2m
5
2m
•Remember < and > use open dots •Remember ≤ and ≥ use closed dots
Solve the inequality and graph the solution set
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Example 2. 7x – 5 > 6x + 4
9
5455
45
)6(46)6(57
4657
x
x
x
xxxx
xx
7 86 1511109 12 13 14
{x| x > 9}
Your Turn
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The solution to an “AND” inequality is the intersection of their graphs (what they share)
10 < 3y-2 < 19
1.6
2 31 10654 7 8 9
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AND Special Cases• x > 5 and x < 1
• No intersection• No Solution
• x > 2 and x > 0
• {x| x > 2}
1 5 -1 20 1
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The solution of an “OR” inequality is the union of their graphs (graph both and keep everything)
x+5>7 or x+2<-2
-5 -4-6 3-1-2-3 0 1 2
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OR Special Cases• x>3 or x<7
All Real Numbers ARN
• x>2 or x > 5
{x | x>2}
3 7 2 5
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Absolute Value Inequalities
•Rules:▫If |a| < b or |a| < b then it is an AND▫If |a| > b or |a| > b the it is an OR
Less thAN ------ AND GreatOR ------ OR
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Example
•|3x-6|<12
54 6-3 21 30-1-2