Transcript Chapter 1
Chapter 1
Equations and Inequalities
1.1
2 Minute Vocabulary Activity
• Take 2 minutes to define and/or give an example
of each vocabulary word below…
Variable
Coefficient
Monomial
Constant
Degree
Order of operations
Term
Like terms
Binomial
Polynomial
Trinomial
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)
1.2
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
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 belongs
a. √6
b. 5
c. -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 2
Name the property
a. ( -8 + 8) + 15 = 0 + 15
b. ( 5 + 7) + 8 = 8 + (5 + 7)
c. ¼ (4x) = x
1.3
Verbal Expressions to Algebraic
Expressions
• 1. Write an algebraic expression to
represent each verbal expression
a. 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
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
a.
a + 4.39 = 76
b. -3d = 18
5
Practice #3
c. 2(2x + 3) – 3(4x – 5) = 22
d. -10x + 3(4x – 2) = 6
Apply the properties of Equality
9
• If 3n 8 what is the value of 3n - 3
5
• If
8 what is the value of 5y - 6
5y 2
3
Solve for a Variable
• The formula for the surface area S of a cone
2
is S rl rwhere
l is the slant height of
the cone and r is the radius of the bas. Solve
the formula for l.
16
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?
1.4
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
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 Equation
c. |x – 18| = 5
d. 9 = |x + 12|
e. 8 = |y + 5|
f. |5x – 6 | + 9 = 0
Solve an Absolute Value Equation
g. |x + 6| = 3x – 2
h. 2|x + 1| - x = 3x – 4
i. -2|3a – 2| = 6
j. 3|2x + 2| - 2x = x + 3
1.5
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
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
Solve the inequality and graph the solution set
• Example 1
m4
m
9
9m m 4
10m 4
4
m
10
2
m
5
-1
0
•Remember < and >
use open dots
•Remember ≤ and ≥
use closed dots
2
m
5
Your Turn
Example 2.
7x – 5 > 6x + 4
7x 5 6x 4
7 x 5 (6 x) 6 x 4 (6 x)
x 5 4
x 55 45
x9
6
7
{x| x > 9}
8
9
10
11
12
13
14
15
1.6
The solution to an “AND” inequality is the
intersection of their graphs (what they
share)
10 < 3y-2 < 19
1
2
3
4
5
6
7
8
9
10
AND Special Cases
• x > 5 and x < 1
1
• No intersection
• No Solution
• x > 2 and x > 0
5
-1
0
• {x| x > 2}
1
2
The solution of an “OR” inequality is the
union of their graphs (graph both and keep
everything)
x+5>7
-6
or x+2<-2
-5
-4
-3
-2
-1
0
1
2
3
OR Special Cases
• x>3 or x<7
3
All Real Numbers
ARN
• x>2 or x > 5
7
2
{x | x>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
-3
-2
-1
0
1
2
3
4
5
6