1. Equations and Inequalities
Chapter 1
Equations and Inequalities

2. 2 Minute Vocabulary Activity
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

3. 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

4. Example 2:
Evaluate 8xy + z3 if x = 5, y = -2, and z = -1
y2 + 5

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)

6. What are the different types of numbers?
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

7. 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

8. Practice Example 1: Name the sets to which each number belongs a. √63

9. 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

10. More Practice Example 2 Name the property a. ( -8 + 8) + 15 = 0 + 15
b. ( 5 + 7) + 8 = 8 + (5 + 7)
c. ¼ (4x) = x

11. Verbal Expressions to Algebraic Expressions
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

12. 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

13. Solving Equations Practice #3
b. -3d = 18 5
c. 2(2x + 3) – 3(4x – 5) = 22
d. -10x + 3(4x – 2) = 6

14. Apply the properties of Equality
If what is the value of 3n - 3
If what is the value of 5y - 6

15. 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.

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?

17. 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

18. a. 1.4 + |5y – 7| if y = -3 b. |4x + 3| - 3 ½ if x = -2
Work in pairs (speed-date activity) Evaluate an Expression with Absolute Value
a |5y – 7| if y = -3
b. |4x + 3| - 3 ½ if x = -2

19. Solve an Absolute Value Equation
c. |x – 18| = 5
d. 9 = |x + 12|
e. 8 = |y + 5|
f. |5x – 6 | + 9 = 0

20. Solve an Absolute Value Equation
g. |x + 6| = 3x – 2
h. 2|x + 1| - x = 3x – 4
i |3a – 2| = 6
j. 3|2x + 2| - 2x = x + 3

21. Remember those Algebra 1 Properties?
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
x < -8

22. Set-Builder Notation- How to write you answers
The solution set of an inequality
Example
-0.25y > 2
-0.25y > 2
y < -8
{y | y < -8}
*read the set of all y such that y is less than or equal to negative 8

23. Solve the inequality and graph the solution set
Example 1
Remember < and > use open dots
Remember ≤ and ≥ use closed dots -1

24. Your Turn Example 2. 7x – 5 > 6x + 4 {x| x > 9} 14 15 6 7 8 9 1011 12 13

25. 1.6
The solution to an “AND” inequality is the intersection of their graphs (what they share)
10 < 3y-2 < 19 9 10 1 2 3 4 5 6 7 8

26. AND Special Cases x > 5 and x < 1 No intersection No Solution
{x| x > 2} 1 5 -1 1 2

27. The solution of an “OR” inequality is the union of their graphs (graph both and keep everything)
x+5>7 or x+2<-2 2 3 -6 -5 -4 -3 -2 -1 1

28. OR Special Cases x>3 or x<7 All Real Numbers ARN
{x | x>2} 3 7 2 5

29. 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

30. Example
|3x-6|<12 -3 -2 5 -1 1 2 3 4 6