Algebra 2 Chapter 1

Expressions and Formulas Section 1.1 Expressions and Formulas

Review of Key Vocabulary Variables: Symbols (letters) used to represent unknown quantities. Algebraic Expressions: Expressions that contain at least one variable. Monomial: An algebraic expression that is a number, variable, or product of a number and one or more variables. Constants: Monomials that contain no variables. Coefficient: The numerical factor of a monomial.

Review of Key Vocabulary Degree: (of a monomial) is the sum of the exponents of its variables. Power: An expression in the form of xn . The word is also used to refer to the exponent itself. Polynomial: A monomial or a sum of monomials. Terms: (of a polynomial) the monomials that make up a polynomial. Like Terms: Monomials that can be combined. The have the same variables to the same powers.

Review of Key Vocabulary Trinomial: A polynomial that has three unlike terms. Binomal: A polynomial that has two unlike terms. Formula: A mathematical sentence that expresses the relationship between certain quantities.

Practice Problems – Evaluating Expressions Evaluate each expression if 𝑥 = 4, 𝑦 = − 2, and 𝑧 = 3.5. 𝑧 – 𝑥 + 𝑦 2. 𝑥 + (𝑦 – 1)3 3. 𝑥 + [3(𝑦 + 𝑧) – 𝑦] 4. 𝑥 2 −𝑦 𝑧+2.5

Practice Problems – Using Formulas Simple interest is calculated using the formula 𝐼=𝑝𝑟𝑡, where p represents the principal in dollars, r represents the annual interest rate, and t represents the time in years. Find the simple interest I given in each set of values. 1. p = $1,800, r = 6%, t = 4 years 2. p = $31,000, r = 2 ½ %, t = 18 months

Properties of Real Numbers Section 1.2 Properties of Real Numbers

R = Reals I = Irrationals W = Wholes Q = Rationals Z = Integers N = Naturals

Practice – Sets of Numbers Name the sets of numbers to which each number belongs: 5 6 − 2 3 – 43 – 23.3

Properties of Real Numbers Property Addition Multiplication Commutative 𝑎+𝑏=𝑏+𝑎 𝑎∗𝑏=𝑏∗𝑎 Associative 𝑎+𝑏 +𝑐=𝑎+(𝑏+𝑐) 𝑎∗𝑏 ∗𝑐=𝑎∗(𝑏∗𝑐) Identity 𝑎+0=𝑎=0+𝑎 𝑎∗1=𝑎=1∗𝑎 Inverse 𝑎+ −𝑎 =𝟎= −𝑎 +𝑎 𝐼𝑓 𝑎 ≠0, 𝑡ℎ𝑒𝑛 𝑎∗ 1 𝑎 =𝟏= 1 𝑎 ∗𝑎 Distributive 𝑎 𝑏+𝑐 =𝑎𝑏+𝑎𝑐 and 𝑏+𝑐 𝑎=𝑏𝑎+𝑐𝑎

Practice – Properties of Real Numbers Name the property illustrated by: −8+8 +15=0+15 Identify the additive inverse and multiplicative inverse for – 7.

Practice – Simplifying Expressions 3 4𝑥−2𝑦 −2(3𝑥+𝑦) Simplify: 1 2 16−4𝑎 − 3 4 (12+20𝑎)

Assignment A#1.10: Page 50 [11-27] Due in class Monday, Sept 9

Section 1.3 Solving Equations

Key Vocabulary Open Sentence: A mathematical sentence containing one or more variables. Equation: A mathematical sentence stating two mathematical expressions are equal. Solution: (of an open sentence) Each replacement of a number for a variable in an open sentence that results in a true sentence.

Properties of Equality Property Symbols Examples Reflexive For any real number a, a = a −7+𝑛=−7+𝑛 Symmetric For all real numbers, a and b, if a = b, then b = a 𝐼𝑓 3=5𝑥+6, 𝑡ℎ𝑒𝑛 5𝑥+6=3 Transitive For all real numbers a, b, and c, if a = b and b = c, then a = c. 𝐼𝑓 2𝑥+1=7 𝑎𝑛𝑑 7=5𝑥−8, 𝑡ℎ𝑒𝑛 2𝑥+1=5𝑥−8. Substitution If a = b, then a may be replaced by b and b may be replaced by a. 𝐼𝑓 4+5 𝑚=18, 𝑡ℎ𝑒𝑛 9𝑚=18.

Practice – Algebraic to Verbal Sentence Write a verbal sentence to represent each equation: 𝑔−5=−2 2𝑐= 𝑐 2 −4

Practice – Properties of Equality Name the property illustrated by each statement: 𝐼𝑓 −11𝑎+2=−3𝑎, 𝑡ℎ𝑒𝑛 −3𝑎=−11𝑎+2. 𝑎−2.03=𝑎−2.03

Tips to Remember When Solving Equations… Goal of solving an equation: Get the variable alone on one side of the equation and everything else on the other side. What you do to one side of the equation, you MUST do to the other side. Checking solutions to discover possible errors is a vital procedure when you use math on the job. Use reverse-PEMDAS when solving multi-step equations.

Practice – Solving Equations Solve: 𝑥−14.29=25 Solve : 2 3 𝑦=−18

Practice – Solving Equations Solve: −10𝑥+3 4𝑥−2 =6 Solve : 2 2x−1 −4 3x+1 =2

Practice – Solving Equations If 5y+2= 8 3 , what is the value of 5𝑦−6?

Practice – Solving Equations The formula for the surface area S of a cylinder is S=2π 𝑟 2 +2𝜋𝑟ℎ, where 𝑟 is the radius of the base, and ℎ is the height of the cylinder. Solve the formula for ℎ

Assignment A#1.30: Page 51 [28-39] Due on Tuesday, Sept 10

Solving Absolute Value Equations Section 1.4 Solving Absolute Value Equations

Absolute Value Absolute value bars act as a grouping symbol!

Evaluate the following: 4𝑥+3 −3 1 2 if 𝑥 = −2 1 1 3 −|2𝑦+1| if 𝑦 =− 2 3

Solve the following: 9= 𝑥+12 8=|𝑦+5|

Note: the Empty Set Because the absolute value of a number is always positive or zero, an equation like 𝑥 =−5 is never true. It, therefore, has no solution. We call this the “empty set”. Represented by either {} or ∅.

Solve the following: −2 3𝑎−2 =6 4𝑏+1 +8=0 2 𝑥+1 −𝑥=3𝑥−4 3 2𝑥+2 −2𝑥=𝑥+3

Assignment A#1.40: Page 51 [40-46] Due:

Section 1.5 Solving Inequalities

Trichotomy Property For any two real numbers, 𝑎 and 𝑏, exactly one of the following statements are true: 𝑎<𝑏 𝑎=𝑏 𝑎>𝑏 Adding the same number to, or subtracting the same number from, each side of an inequality does NOT change the truth of the inequality.

Properties of Inequality Note: the properties are also true for ≤,≥,≠ Addition Property of Inequality Words Example For any real numbers, 𝑎, 𝑏, and 𝑐: If 𝑎>𝑏, then 𝑎+𝑐>𝑏+𝑐. If 𝑎<𝑏, then 𝑎+𝑐<𝑏+𝑐. 3<5 3+ −4 <5+ −4 −1<1 Subtraction Property of Inequality Words Example For any real numbers, 𝑎, 𝑏, and 𝑐: If 𝑎>𝑏, then 𝑎−𝑐>𝑏−𝑐. If 𝑎<𝑏, then 𝑎−𝑐<𝑏−𝑐. 2>7 2−8>7−8 −6>−15

Practice – Solve an Inequality Using Addition Solve 4𝑥+7≤3𝑥+9. Graph the solution set on a number line.

Multiplication Property of Inequality Note: the properties are also true for ≤,≥,≠ Multiplication Property of Inequality Words Examples For any real numbers, a, b, and c, where: c is positive: If 𝑎>𝑏, the 𝑎𝑐>𝑏𝑐 −2<3 4 −2 <4 3 −8<12 If 𝑎<𝑏, then 𝑎𝑐<𝑏𝑐 c is negative: If 𝑎>𝑏, then 𝑎𝑐<𝑏𝑐 5>−1 −3 5 < −3 −1 −15<3 If 𝑎<𝑏, then 𝑎𝑐>𝑏𝑐

Division Property of Inequality Note: the properties are also true for ≤,≥,≠ Division Property of Inequality Words Examples For any real numbers, a, b, and c, where: c is positive: If 𝑎>𝑏, then 𝑎 𝑐 > 𝑏 𝑐 −18<−9 −18 3 < −9 3 −6<−3 If 𝑎<𝑏, then 𝑎 𝑐 < 𝑏 𝑐 c is negative: If 𝑎>𝑏, then 𝑎 𝑐 < 𝑏 𝑐 12>8 12 −2 < 8 −2 −6<−4 If 𝑎<𝑏, then 𝑎 𝑐 > 𝑏 𝑐

Set-builder Notation 𝑥 𝑥>9 is read “The set of all numbers x such that x is greater than 9.” { } (called braces) denotes “the set of” | denotes “such that”

Practice – Solve an Inequality Using Multiplication Solve − 1 3 𝑥<1. Graph the solution set on a number line.

Solve a Multi-Step Inequality Solve 3 2𝑞−4 >6. Graph the solution set on a number line.

Solve a Multi-Step Inequality Solve −𝑥> 𝑥−7 2 . Graph the solution set on a number line.

Assignment A#1.50: Page 52 [47-53] Due:

Solving Compound and Absolute Value Inequalities Section 1.6 Solving Compound and Absolute Value Inequalities

Compound Inequality Consists of two inequalities joined by the word and (a conjunction) or the word or (a disjunction). To solve, you must solve each part. Compound inequality graph: see intersection on page 41 in your text.

Solve an “and” Compound Inequality Solve and graph the solution set on a number line. 10≤3𝑦−2<19

Solve an “or” Compound Inequality Solve and graph the solution set on a number line. 𝑥+3<2 𝑜𝑟 −𝑥≤−4

Solve an Absolute Value Inequality (<) Solve and graph the solution set on a number line. |𝑥|≤3

Solve an Absolute Value Inequality (>) Solve and graph the solution set on a number line. 𝑥 ≥3

Absolute Value Inequalities For all real numbers a and b, b > 0, the following statements are true: 𝐼𝑓 𝑎 <𝑏, 𝑡ℎ𝑒𝑛 −𝑏<𝑎<𝑏. 𝐼𝑓 2𝑥+1 <5, 𝑡ℎ𝑒𝑛 −5<2𝑥+1<5. 𝐼𝑓 𝑎 >𝑏, 𝑡ℎ𝑒𝑛 𝑎>𝑏 𝑜𝑟 𝑎<−𝑏. 𝐼𝑓 2𝑥+1 >5, 𝑡ℎ𝑒𝑛 2𝑥+1>5 𝑜𝑟 2𝑥−1<−5.

Solve a Multi-Step Absolute Value Inequality Solve and graph the solution set on a number line. 3𝑥+4 <10 |2𝑥−2|≥4

Assignment A#1.60: Page 52 [54-62] and Page 49 [1-10] Due: