Equations and Inequalities Chapter 1 Equations and Inequalities

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

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? 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 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

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

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 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

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.

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

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. 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

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 -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 Remember < and > use open dots Remember ≤ and ≥ use closed dots -1

Your Turn Example 2. 7x – 5 > 6x + 4 {x| x > 9} 14 15 6 7 8 9 10 11 12 13

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

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

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

OR Special Cases x>3 or x<7 All Real Numbers ARN {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 -3 -2 5 -1 1 2 3 4 6