Chapter 1 Equations and Inequalities. 2 Minute Vocabulary Activity Take 2 minutes to define and/or...
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Transcript of Chapter 1 Equations and Inequalities. 2 Minute Vocabulary Activity Take 2 minutes to define and/or...
Chapter 1Equations and Inequalities
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
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
•Example 2:▫Evaluate 8xy + z3 if x = 5, y = -2, and z =
-1 y2 + 5
•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)
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
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
Practice
Example 1:Name the sets to which each number
belongsa. √6b. 5c. -2
3
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
More Practice
•Example 2Name the property
a. ( -8 + 8) + 15 = 0 + 15
b. ( 5 + 7) + 8 = 8 + (5 + 7)
c. ¼ (4x) = x
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
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
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
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
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
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
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
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
Solve an Absolute Value Equationc. |x – 18| = 5
d. 9 = |x + 12|
e. 8 = |y + 5|
f. |5x – 6 | + 9 = 0
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
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
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
•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
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
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
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
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
OR Special Cases• x>3 or x<7
All Real Numbers ARN
• x>2 or x > 5
{x | x>2}
3 7 2 5
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
Example
•|3x-6|<12
54 6-3 21 30-1-2